representation theory

representation theory (countable and uncountable, plural representation theories)

1989 [Clarendon Press], Robert J. Baston, Michael G. Eastwood, The Penrose Transform: Its Interaction with Representation Theory, 2016 Dover edition.

1998, A. Broer, A. Daigneault, Representation Theories and Algebraic Geometry, Springer:

2005, Andrea Brini, Francesco Regonati, Antonio Teolis, Combinatorics and Representation Theory of Lie Superalgebras over Letterplace Superalgebras, Hongbo Li, Peter J. Olver, Gerald Sommer (editors), Computer Algebra and Geometric Algebra with Applications: 6th International Workshop IWWM and International Workshop GIAE, Springer, LNCS 3519, page 239,
As an application, we describe in detail the way to specialize general results to the representation theory of the symmetric group and to classical invariant theory.

From his numerous contributions through research articles, books, and conference volumes we just mention his^([Vlastimil Dlab's]) work on the representation theory of hereditary algebras in the mid-seventies of the last century, joint with C.M. Ringel, laying the foundations of, in particular, tame hereditary representation theory over arbitrary fields, a theory still at the centre of interest.

2005, Ragnar-Olaf Buchweitz, Helmut Lenzing, Representations of Algebras and Related Topics, American Mathematical Society, page ix:

linear transformation
linear transformation

(mathematics, algebra) A branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.


free Boolean algebra

en-noun

free Boolean algebra (plural free Boolean algebras)

(algebra) A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas). Starting with a set of n variables which are independent of each other and are called generators, the power set of this set has 2ⁿmembers which may be called atoms and are valuations of the n variables: a valuation can be considered to be a set of variables which are "true" under that valuation, or a conjunction of generators (such that variables not included in that set are included in negated form in the equivalent conjunction). Then the power set of the set of atoms yields a set of 2^(2ⁿ) members which are the elements of the said field of sets. These elements correspond to Boolean formulas: a formula can be considered to be a set of valuations which make the formula true, or a linear combination (i.e., a disjunction) of atoms.


free abelian group

free abelian group (plural free abelian groups)

A free abelian group of rank n is isomorphic to ℤ⊕ℤ⊕...⊕ℤ=⨁ⁿℤ, where the ring of integers ℤ occurs n times as the summand. The rank of a free abelian group is the cardinality of its basis. The basis of a free abelian group is a subset of it such that any element of it can be expressed as a finite linear combination of elements of such basis, with the coefficients being integers. (For an element a of a free abelian group, 1a = a, 2a = a + a, 3a = a + a + a, etc., and 0a = 0, (−1)a = −a, (−2)a = −a + −a, (−3)a = −a + −a + −a, etc.)

(algebra) A free module over the ring of integers.


cyclotomic polynomial

cyclotomic polynomial (plural cyclotomic polynomials)

For a prime number p, the pᵗʰ cyclotomic polynomial is #123;xᵖ-1#92;overx-1#125;#61;x#123;p-1#125;#43;x#123;p-2#125;#43;...#43;x²#43;x#43;1.

Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x#43;1.

Euler's totient function
Euler's totient function

(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial.


prime ideal

By analogy with the notion of prime number in number theory.

Given a prime number p, there is only a finite number of prime ideals #92;mathfrak#123;p#125; in #92;mathfrak#123;o#125; such that #92;mathfrak#123;p#125;#92;capJ#61;p (they are the prime ideals of #92;mathfrak#123;o#125;p).

1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39:

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189,
In the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o on the basis of Axiom II; thus, in that section there are no lower prime ideals but o. Since every ideal a ne o is divisible by a prime ideal distinct from o (proof: from among all the divisors of a distinct from o choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o.

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:

(algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.


free monoid

(algebra) A monoid whose underlying set is the Kleene closure of some set of generators, and whose operator is concatenation.


quotient ring

The third section covers a construct similar to the ring S°R — the maximal quotient ring, which exists for any ring. (When R is nonsingular, the maximal quotient ring is exactly S°R.) Finally, Section D provides an answer to the question of which right and left nonsingular rings have coinciding maximal right and left quotient rings.

1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39:

On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.

2006, Peter A. Linnell, “Noncommutative localization in group rings”, in Andrew Ranicki, editor, Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42:

2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain.

2012, Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, translated by Svetla Nikova, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10:

(algebra, ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R.

anneau quotient

ring whose elements are the cosets of an ideal


fractional ideal

fractional ideal (plural fractional ideals)

Products of fractional ideals are again fractional ideals, since if A#92;alpha#92;subseteqR and B#92;beta#92;subseteqR, then (AB)(#92;alpha#92;beta)#92;subseteqR.

1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476:

Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group.

2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10:

2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106,
A fractional ideal of F is a finitely generated 𝒪_F-submodule of F. Fractional ideals form a group together with the ideal multiplication. The neutral element is 𝒪_F and the inverse of a fractional ideal a⊂F is
a⁻¹=x∈F;xa⊂𝒪_F.
[…] Two fractional ideals a,b are called equivalent, if there is a r∈F such that a=rb.

(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.

R-submodule of Frac(R) such that for some nonzero r∈R


field of fractions

field of fractions (plural fields of fractions)

1971 [Wadsworth Publishing], Allan Clark, Elements of Abstract Algebra, 1984, Dover, page 175,
The general construction of the field of fractions ℚ_R out of R is an exact parallel of the construction of the field of rational numbers ℚ out of the ring of integers ℤ.

1989, Nicolas Bourbaki, Commutative Algebra: Chapters 1-7, [1985, Éléments de Mathématique Algèbre Commutative, 1-4 et 5-7, Masson], Springer, page 535,
In this no., A and B denote two integrally closed Noetherian domains such that A ⊂ B and B is a finitely generated A-module and K and L the fields of fractions of A and B respectively.

We denote by #92;mathcal#123;A#125;(#92;mathbb#123;K#125;) the #92;mathbb#123;K#125;-algebra of entire functions in #92;mathbb#123;K#125; i.e. the set of power series with coefficients in #92;mathbb#123;K#125; converging in all #92;mathbb#123;K#125; and we denote by #92;mathcal#123;M#125;(#92;mathbb#123;K#125;) the field of meromorphic functions in #92;mathbb#123;K#125;, i.e. the field of fractions of #92;mathcal#123;A#125;(#92;mathbb#123;K#125;).

2013, Jean-Paul Bézivin, Kamal Boussaf, Alain Escassut, “Some old and new results on the zeros of the derivative of a p-adic meromorphic function”, in Khodr Shamseddine, editor, Advances in Ultrametric Analysis: 12th International Conference on p-adic Functional Analysis, American Mathematical Society, page 23:

(algebra, ring theory) The smallest field in which a given ring can be embedded.


factor ideal

1965 [Prentice-Hall], Seth Warner, Modern Algebra, 1990, Dover, page 651,
Consequently, we may speak of the sequence of invariant factor ideals of a finitely generated unitary module over a principal ideal domain A.

1985 [W. H. Freeman and Company], Nathan Jacobson, Basic Algebra I, 2nd Edition, Dover, 2009, page 193,
We shall now call the sequence of order ideals, ann z₁, ann z₂, ..., whose uniqueness has just been proved, the invariant factor ideals of the module M.

Recall that a factor ideal I of a half-shell A comes from a factor congruence θ of A where I = 0/θ. Factor ideals and congruences will be needed whenever we want to show that we have embedded a half-shell into a Baer–Stone half-shell.[…]
2.1 Definition. Remember that a half-shell A is called a Baer–Stone half-shell if the right and left annihilators of every single element of A are factor ideals. It is called finitely Baer-Stone if the annihilators on both sides of any finite subset of A are factor ideals.

2012, Arthur Knoebel, Sheaves of Algebras over Boolean Spaces, Springer (Birkhäeser), page 246:

(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R).


factor out

en-verb

factor out (third-person singular simple present factors out, present participle factoring out, simple past and past participle factored out)

factors out (present singular third-person)
factoring out (participle present)
factored out (participle past)
factored out (past)

(transitive, algebra) To isolate a common factor from an expression.

ottaa yhteinen tekijä

(algebra) to isolate a common factor from an expression


simple ring

Theorem 2. A semi-simple ring #92;mathfrak#123;A#125; satisfying the minimum condition can be decomposed in only one way as a direct sum of ideals which are simple rings.

1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page 43:

By theorem 7.7.1 any field is a simple ring.

1969, Frederick Michael Hall, An Introduction to Abstract Algebra, volume 2, Cambridge University Press, page 195:

A field is clearly a simple ring. Indeed, a commutative simple ring with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring is Fₙ, the n × n matrix ring over a field F, n > 1.

1994, P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, page 204:

Since M#95;n(D) has no nonzero proper two-sided ideals, it follows from the above discussion that M#95;n(D) is a left simple ring. We shall see that every simple ring is isomorphic to M#95;n(D) for some n, and for some division ring D.

2017, Ramji Lal, Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group Extensions and Schur Multiplier, Springer, page 335:

(algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself).

anneau simple

ring that contains no ideals other than the zero ideal and the ring itself


simple algebra

1945 [McGraw-Hill], E. T. Bell, 2nd Edition, 1992, Dover, The Development of Mathematics, page 252,
A semi-simple algebra over F is either simple or the direct sum of simple algebras over F.

1989 [W. H Freeman & Company], Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215,
If A is a finite dimensional simple algebra over F, then A≅M_n(Δ) where Δ is a finite dimensional division algebra over Δ.

One proves that if #92;mathcal#123;L#125; is a simple algebra over #92;mathbb#123;R#125;, then L#95;C is either a simple algebra over #92;mathbb#123;C#125;, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over #92;mathbb#123;R#125; reduces to a similar problem over #92;mathbb#123;C#125;.

2005, Igor R. Shafarevich, translated by Miles Reid, Basic Notions of Algebra, Springer, page 198:

(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0).

algebra semplice

algebra that contains no nontrivial proper ideals and whose multiplication operation is not zero


prime ring

A ring is called a prime ring if the product of nonzero ideals in it remains nonzero. It is obvious that a prime ring is necessarily semi-prime.

1969, Taita Journal of Mathematics, volumes 1-2, page 56:

The ring R is said to be prime if for all nonzero ideals A, B of R we have AB≠0. An ideal P of R is called a prime ideal if R/P is a prime ring. Prime rings and prime ideals are important building blocks in noncommutative ring theory.

1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 223:

The so-called extended centroid of a prime ring, i.e., a field defined as the center of the Martindale ring of quotients, will enable us to extend a part of the theory of central simple algebras to general prime rings.

2014, Matej Brešar, Introduction to Noncommutative Algebra, Springer, page 163:

(algebra, ring theory) Any nonzero ring R such that for any two (two-sided) ideals P and Q in R, the product PQ = 0 (the zero ideal) if and only if P = 0 or Q = 0.

Moreover, the image of φ_A is the smallest subring of A, in the sense that it is contained in any subring of A, and it is called the prime ring of A.

2012, Sebastian Xambo-Descamps, Block Error-Correcting Codes: A Computational Primer, Springer Science & Business Media, page 110:

(algebra, ring theory) Synonym of prime subring.

The image of ℤ or ℤ/mℤ, respectively, in R as described in the above proposition obviously consists of all sums n · 1 in R, where n ∈ ℤ. It is also called the prime ring of R. R itself is called a prime ring if it equals its own prime ring. If p is a prime number, then the field ℤ/pℤ is a also called the prime field of characteristic p.

2012, Thomas Becker, Volker Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer Science & Business Media, page 50:

Theorem 3.6.3. If R is a prime ring of characteristic zero then R is isomorphic to ℤ. If R is a prime ring of characteristic n then R is isomorphic to ℤₙ.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra: From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory, JHU Press, page 81:

(algebra, ring theory, uncommon) A ring which is equal to its own prime subring.


inner automorphism

en-noun

inner automorphism (plural inner automorphisms)

(algebra) An automorphism which is a conjugation (by a fixed element).


outer automorphism

outer automorphism (plural outer automorphisms)

(algebra) An automorphism which is not an inner automorphism.


maximal ideal

Denote the minimal prime ideal in (b) by J#95;#92;mathsf#123;max#125;(#92;lambda), the unique maximal ideal of infinitesimal character #92;lambda.

1994, William M. McGovern, Completely Prime Maximal Ideals and Quantization, American Mathematical Society, page 15:

2004, Ayman Badawi, Abstract Algebra Manual: Problems and Solutions, Nova Science, 2nd Edition, page 87,
Let S be the set of all prime ideals of B, and H be the set of all maximal ideals of B.

In particular, a prime ideal #92;mathfrak#123;p#125;#92;subsetA is a closed point of #92;mathrm#123;Spec#125;#92;A if and only if #92;mathfrak#123;p#125; is a maximal ideal.

2013, Igor R. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, 3rd edition, Springer, page 11:

(algebra, ring theory) An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).


primary ideal

The prime and primary ideals play roles which are (very roughly) similar to those played by prime numbers and by prime.power numbers in elementary arithmetic.

1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10:

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, page 189,
Thus all higher primary ideals are symbolic powers of higher prime ideals.
Prüfer has called the ideals a with the property a* = a v-ideals. The integral v-ideals are just those in whose primary ideal decomposition only higher primary ideals occur.

A primary ideal has a prime ideal as radical, so its corresponding algebraic set is irreducible. Primary ideals can, however, have multiplicity, so they give a finer description of the solution set.

1997, Ralf Fröberg, An Introduction to Gröbner Bases, John Wiley & Sons, page 71:

(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.


conjugacy class

conjugacy class (plural conjugacy classes)

If a group acts on itself through conjugation, then its orbits are its conjugacy classes and its stabilizer subgroups are its centralizers.

equivalence relation
equivalence relation

(algebra) A subset of a group which is an equivalence class in the quotient set of the group divided by conjugation as equivalence relation.


minimal ideal

Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.

1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page vii:

Theorem 12. A semisimple ring R has only a finite number of minimal ideals and is their direct sum. Moreover, each minimal ideal is a simple ring.

1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33:

Note that if a semigroup has a zero, then its minimal ideal is {0}. In this context, the notion of minimal ideal is not so useful, and so we introduce the notion of a 0-minimal ideal.

2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596:

(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.


radical ideal

1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,
Any intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime.

It follows at once that R#47;I is a reduced ring iff I is a radical ideal. Thus, the ideals I(X) are all radical ideals.

1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33:

1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,
Thus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals […] .

(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I).

commutative algebra; ring theory; commutative algebra; ring theory


reduced ring

We do this for reduced rings in Corollary 10, and for rings with nonzero nilpotents in Corollary 15.

1997, Thomas G. Lucas, “Characterizing When R(X) is Completely Integrally Closed”, in Daniel Anderson, editor, Factorization in Integral Domains, Marcel Dekker, page 401:

Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m#92;inM and a#92;inR, ma#61;0 implies mR#92;capMa#61;0. Various results of reduced rings are extended to reduced modules.

2004, Tsiu-Kwen Lee, Yiqiang Zhou, “Reduced Modules”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, Marcel Dekker, page 365:

In general, the first case of importance is the normalization of a reduced ring R in its quotient ring K(R).

2005, David Eisenbud, The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry, Springer, page 210:

(algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0.

reduzierter Ring

ring that has no nonzero nilpotent elements


reduced algebra

reduced algebra (plural reduced algebras)

(algebra, ring theory) An algebra over a reduced ring.


bounded lattice

bounded lattice (plural bounded lattices)

2004, Anna Maria Radzikowska, Etienne Kerre, On L-Fuzzy Rough Sets, Leszek Rutkowski, Jörg Siekmann, Ryszard Tadeusiewicz, Lotfi A. Zadeh (editors), Artificial Intelligence and Soft Computing — ICAISC 2004: 7th International Conference, Proceedings, Springer, LNAI 3070, page 526,
A residuated lattice is an extension of a bounded lattice by a monoid operation and its residuum, which are abstract counterparts of a triangular norm and a fuzzy residual implication, respectively.

2006, Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne Kerre, Triangle Lattices: Towards an Axiomatization of Interval-Valued Residuated Lattices, Salavatore Greco, Yukata Hata, Shoji Hirano, Masahiro Inuiguchi, Sadaaki Miyamoto, Hung Son Nguyen, Roman Słowiński (editors), Rough Sets and Current Trends in Computing: 5th International Conference, Proceedings, Springer, LNAI 4259, page 117,
Indeed, in the scope of these logics, formulas can be assigned not only 0 and 1 as truth values, but also elements of [0,1], or, more generally, of a bounded lattice ℒ.

In this paper, we study idempotent nullnorms on bounded lattices. We prove that there is no idempotent nullnorm on a distributive bounded lattice L different from the proposal in [13].

2018, Gül Deniz Çayli, Funda Karaçal, “Some Remarks on Idempotent Nullnorms on Bounded Lattices”, in Vicenç Torra, Radko Mesiar, Bernard De Baets, editors, Aggregation Functions in Theory and in Practice, Springer,, page 32:

partially ordered set
partially ordered set

(algebra, order theory) Any lattice (type of partially ordered set) that has both a greatest and a least element.


field extension

field extension (plural field extensions)

A Galois field extension may be defined in terms of its Galois group (Section 2) or in terms of the internal structure of the extension (Section 3).

1974, Thomas W. Hungerford, Algebra, Springer, page 230:

1998, David Goss, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
Note that the extension of L obtained by adjoining all division points of ψ includes at most a finite constant field extension.

2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.

(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.

kuntalaajennus

pair of fields such that one is a subfield of the other


prime field

1968, J. F. Schuh, Mathematical Tools for Modern Physics, Philips Technical Library, page 91,
Theorem 2. Each field contains a unique prime field.

2004, H. Gopalkrishna Gadiyar, K. M. Sangeeta Maini, R. Padma, Cryptography, Connections, Cocycles and Crystals: A p-adic Exploration of the Discrete Logarithm Problem, Anne Canteaut, Kapaleeswaran Viswanathan (editors), Progress in Cryptology - INDOCRYPT 2004: 5th International Conference, Proceedings, Springer, LNCS 3348, page 305,
Applying Hensel's lemma to the discrete logarithm problem over prime fields reveals the rich geometric and algebraic structure underlying the problem.

Since every field contains a prime field, by studying prime fields, and the way properties are passed over to extensions, Steinitz would attain a full picture of the structure of all possible fields.

2004, Leo Corry, Modern Algebra and the Rise of Mathematical Structures, 2nd edition, Springer (Birkhäuser), page 194:

(algebra, field theory) A field that contains no proper subfields.


algebraic extension

algebraic extension (plural algebraic extensions)

What we now have to prove is that: if an overring R#42; of R is such that R#42; is an integral domain, R#42; is integral over R and the quotient field #92;mathfrak#123;L#125;#42; of R#42; is a finite algebraic extension of #92;mathfrak#123;L#125;, then R#42; is a local ring and a finite R-module.

1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200:

It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.

2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110:

We describe the correspondence between subfields of the field P#95;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #92;pi#95;1(X#92;setminusO).

2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65:

(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K).


discrete valuation

discrete valuation (plural discrete valuations)

Therefore the theory branches in two essential parts : first the study of discrete valuations of the Weyl field D#95;1(K) extending a discrete valuation of K, secondly the discrete valuations of D#95;1(K) having the purely transcendental field K(t) as a residue field.

2000, Freddy Van Oystaeyen, Algebraic Geometry for Associative Algebras, CRC Press, page 173:

One of the few possibilities to extend some arithmetic aspects of number theory to more abstract fields is via valuation theory and, as a first step, we may restrict attention to discrete valuations.

2004, Farahat S. Aly, Freddy Van Oystaeyen, “Hopf Order Module Algebra Orders”, in Stefaan Caenepeel, Fred Van Oystaeyen, editors, Hopf Algebras in Noncommutative Geometry and Physics, CRC Press, page 21:

Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field.

2011, Ricardo Portilla, “Parametrizing nilpotent orbits in p-adic symmetric spaces”, in Robert S. Doran, Paul J. Sally, Jr., Loren Spice, editors, Harmonic Analysis on Reductive, p-adic Groups, American Mathematical Society, page 201:

(algebra, field theory) A valuation (on some field) that takes integer values (including infinity).


combinatorial commutative algebra

combinatorial commutative algebra (uncountable)

This construction has become a fundamental tool in combinatorial commutative algebra due to the work of Stanley, Hochster and Reisner ([Ho77], [S75], [S96], [R76]).

2011, Viviana Ene, Jũrgen Herzog, Gröbner Bases in Commutative Algebra, American Mathematical Society, page 88:

2013, Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón Introduction, Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón (editors), Monomial Ideals, Computations and Applications, Springer, page vii,
Monomial ideals and algebras are among the simplest structures in commutative algebra and the main objects in combinatorial commutative algebra.

Using techniques from combinatorial commutative algebra [MS05, KM], they showed that certain lattice games admit a particular type of algebraic decomposition known as affine stratifications, from which a finitely representable winning strategy can be recovered.

2013, Aaron N. Siegel, Combinatorial Game Theory, American Mathematical Society, page 269:

(algebra) A relatively new discipline in mathematics that combines techniques and concepts from combinatorics and commutative algebra, and in which the geometry of convex polytopes also plays a significant role.


zero divisor

zero divisors (plural)
zero-divisor (alternative)

regular element

antonym(s) of “any element whose product with some nonzero element is zero”

trivial zero divisor

An idempotent element e#92;ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)#61;0#61;(1-e)e.

Linnell [25, 1977] proved that if G is a torsion-free abelian by locally finite by super-solvable group and K is any field, then K[G] has no nontrivial zero divisors.

1984, J. B. Srivastava, “23: Projective Modules, Zero Divisors, and Noetherian Group Algebras”, in Dinesh N. Manocha, editor, Algebra and its Applications, CRC Press, page 170:

In the ring of integers, there are no zero divisors except 0. In a ring obtained from a Boolean algebra, on the other hand, every element except the identity is a zero-divisor.
The concept of a zero-divisor is intimately related to cancellation law as we see n the following proposition.
1.7 Proposition: Let R be a ring and x#92;inR. Then for all y,x#92;inR, either of the equations xy#61;xz or yx#61;zx implies y#61;z if and only if x is not a zero divisor. In other words, cancellation by an element is possible iff it is not a zero-divisor.

1989, K. D. Joshi, Foundations of Discrete Mathematics, New Age International, page 390:

In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors.

2010, Mitsuo Kanemitsu, “The Number of Distinct 4-Cycles and 2-Matchings of Some Zero Divisor Graphs”, in Masami Ito, Yuji Kobayashi, Kunitaka Shoji, editors, Automata, Formal Languages and Algebraic Systems: Proceedings of AFLAS 2008, World Scientific, page 63:

(algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

If R is an integral domain, that is, has no zero divisors, then R#91;x#93; also has no zero divisors.

2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234:

An element a of a ring is called a zero-divisor if a#92;ne 0 and ab#61;0 or ba#61;0 for some b#92;ne 0; if a is neither 0 nor a zero-divisor, it is said to be regular (see Section 7.1). A non-trivial ring without zero-divisors is called an integral domain; this term is not taken to imply commutativity.

2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi:

If a and b are non-zero elements of R such that ab#61;0, then aandb are both called zero divisors. If R is non-trivial and has no zero divisors, then it is called an integral domain. Note that if a is a unit in R, it cannot be a zero divisor (if ab#61;0, then multiplying both sides of this equation by a#123;-1#125; yields b#61;0.

2009, Victor Shoup, A Computational Introduction to Number Theory and Algebra, 2nd edition, Cambridge University Press, page 171:

(algebra, ring theory) A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.


ordered ring

1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
If < is an ordering on A compatible with its ring structure, we shall say that (A,+,·,<) is an ordered ring. An element x of an ordered ring A is positive if x>0, and x is strictly positive if x>0.
The set of all positive elements of an ordered ring A is denoted by A_+, and the set of all strictly positive elements of A is denoted by A^*₊.
If (A,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.

(OR) The relations x>0 and y>0 imply xy>0.
The ring A, together with such an ordering, is called an ordered ring.
Examples. — 1) The rings Q and Z , with the usual orderings, are ordered rings.
2) A product of ordered rings, equipped with the product ordering, is an ordered ring. In particular, the ring Aᴱ of mappings from a set E to an ordered ring A is an ordered ring.
3) A subring of an ordered ring, with the induced ordering, is an ordered ring.

1990, P. M. Cohn, J. Howie (translators), Nicolas Bourbaki, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19, DEFINITION 1. — Given a commutative ring A, we say that an ordering on A is compatible with the ring structure on A if it is compatible with the additive group structure of A, and if it satisfies the following axiom

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4gt;2 because 4-2gt;0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b#92;inR, then we know by trichotomy that exactly one of the following must be true: a-bgt;0, a-b#61;0, or -(a-b)gt;0.

2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253:

(1) The set R⁺ is closed under addition and multiplication.
(2) If x∈R then exactly one of the following is true: (trichotomy law)
(a) x=0,
(b) x∈R⁺,
(c) -x∈R⁺.
If further R is an integral domain we call R an ordered integral domain.
[…]
Lemma 3.5.9. If R is an ordered ring and a∈R is a positive element, then the set na:n∈ N⊂R⁺.
[…]
Theorem 3.5.2. An ordered ring must be infinite.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Johns Hopkins University Press, page 77, Definition 3.5.4. A ring R is an ordered ring if there exists a distinguished set R⁺, R⁺⊂R, called the set of positive elements, with the properties that

(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.


invariant theory

invariant theory (countable and uncountable, plural invariant theories)

1993, Bernd Sturmfels, Introduction, David Hilbert, Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, Cambridge University Press, page xi,
Today, invariant theory is often understood as a branch of representation theory, algebraic geometry, commutative algebra, and algebraic combinatorics. Each of these four disciplines has roots in nineteenth-century invariant theory. […] In modern terms, the basic problem of invariant theory can be categorized as follows. Let V be a K-vector space on which a group G acts linearly. In the ring of polynomial functions K[V] consider the subring K[V]ᴳ consisting of all polynomial functions on V which are invariant under the action of the group G. The basic problem is to describe the invariant ring K[V]ᴳ. In particular, we would like to know whether K[V]ᴳ is finitely generated as a K-algebra and, if so, to give an algorithm for computing generators.

Invariant theory is the great romantic story of mathematics.[…]In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are all offsprings of invariant theory.

2001, Gian-Carlo Rota, “What is invariant theory, really?”, in H. Crapo, D. Senato, editors, Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota, Springer,, page 41:

For a linear algebraic group G and a regular representation (#92;rho,V) of G, the basic problem of invariant theory is to describe the G-invariant elements (#123;#92;bigotimes#125;ᵏⱽ)ᴳ of the k-fold tensor product for all k.

2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 225:

(algebra, representation theory) The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.


Hurwitz algebra

Hurwitz algebra (plural Hurwitz algebras)

Named after German mathematician Adolf Hurwitz (1859–1919); see also Hurwitz problem, Hurwitz's theorem.

As we recall, a Hurwitz algebra is a composition algebra with unit element.

1995, Susumo Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, page 40:

Observe that the unital composition algebra associated with (C,#92;star) by the construction given in the proof of Proposition (33.27) is the Hurwitz algebra (C,#92;diamond) if we set a#61;1.

1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 465:

2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 15,
Unital composition algebras (or Hurwitz algebras) form a well-known class of algebras. Any Hurwitz algebra has a finite dimension equal to either 1, 2, 4, or 8. The two-dimensional Hurwitz algebras are the quadratic étale algebras over the ground field F, the four dimensional ones are the generalized quaternion algebras, and the eight dimensional Hurwitz algebras are called Cayley algebras, and are analog to the classical algebra of octonions (for a good survey of the latter, see [Bae02]).

(algebra) Any one of the unital composition algebras identified by Hurwitz's theorem (on composition algebras) as solutions to the Hurwitz problem.


integral element

integral element (plural integral elements)

1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, American Mathematical Society Translations, Series 2, Volume 3, page 21,
A subring of 𝔅 containing the ring o of integral elements of the field k_0(π), distinct from 𝔅, and not contained in any other subring of 𝔅 distinct from 𝔅, is called a maximal ring of the algebra 𝔅. In a division algebra, the only maximal ring is the ring of integral elements.

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,
If 𝔖 is the ring of integral elements in a commutative ring 𝔗 (over a subring R) and if the element t of 𝔗 is integral over 𝔖, then t is also integral over R (that is, contained in 𝔖).

In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^([Richard Dedekind's]) theory.

2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:

(algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R.

elemento intero

element of a given ring that is a root of some monic polynomial with coefficients in a given subring


algebraic poset

algebraic poset (plural algebraic posets)

1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
A poset (P,<) is said to be algebraic if and only if
i) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists,
ii) for every x∈P, the set
K_X:=y∈P|ycompact,y<x
is non-empty and up-directed, and
x= operatorname supK_x.
A poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<y for some compact element c of P.
Concerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets K_x fail to be up-directed ([50], 4.2 or [49], 4.5). Even when "enough" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets K_x.
The concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) "algebraic lattice" (cf. [9], [20], I-4).

ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions.
A structure (D,#92;le#95;D,#92;cup#95;D) where (D,#92;le#95;D) is an algebraic poset and #92;cup#95;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#92;cup#95;Dd#61;d for all d#92;inD) will be called a nondeterministic algebraic poset.

1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:

It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #123;A#42;#92;cupA#42;#125;#92;sqrt#123;#125; in the prefix order; this poset enjoys the added condition that every element is compact.

1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:

The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.

1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:

(algebra, order theory) A partially ordered set (poset) in which every element is the supremum of the compact elements below it.


polynomial ring

polynomial ring (plural polynomial rings)

It then follows that if A is a graded ring over a local ring, A is a homomorphic image of a polynomial ring over a regular local ring. For the sake of brevity, we refer to a graded polynomial ring over a regular local ring simply as a graded polynomial ring.

1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270:

In Section 3.2 we shall study the special properties of a polynomial ring over a field; for the moment we note a property of polynomial rings which applies quite generally, the Hilbert basis theorem (after David Hilbert, 1862-1943):
Theorem 3.3
If R is any right Noetherian ring, the polynomial ring R#91;x#93; is again right Noetherian.

2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106:

Because they possess a rich theory and provide an excellent source of examples and counterexamples, integer-valued polynomial rings have attained some prominence in the theory of non-Noetherian commutative rings.

2009, Jesse Elliott, “Some new approaches to integer-valued polynomial rings”, in Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson, editors, Commutative Algebra and Its Applications: Proceedings of the 5th International Fez Conference, Walter de Gruyter, page 223:

(algebra) A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K.


composition algebra

composition algebra (plural composition algebras)

1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11,
More precisely, Xⁿ⊂ℙᴺ is a Severi variety if and only if ℙᴺ=ℙ(𝔍), where 𝔍 is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra 𝔄, and X corresponds to the cone of Hermitian matrices of rank <1 (in that case SX corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, X is a Severi variety if and only if X is the “Veronese surface” over one of the composition algebras over the field K (Theorem 4.9).

We call a composition algebra with an associative norm a symmetric composition algebra and denote the full subcategory of #92;mathsf#123;Comp#125;#95;m consisting of symmetric composition algebras by #92;mathsf#123;Scomp#125;#95;m.

1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 464:

2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 150,
At least in the split cases, this is a construction that depends on two unital composition algebras, since the Jordan algebra involved consists of the 3 x 3-hermitian matrices over a unital composition algebra.

(algebra) A non-associative (not necessarily associative) algebra, A, over some field, together with a nondegenerate quadratic form, N, such that N(xy) = N(x)N(y) for all x, y ∈ A.


primitive element

primitive element (plural primitive elements)

primitiivinen alkio

element of a coalgebra satisfying a particular condition

An algebraic extension L#47;K is called simple if L#61;K(#92;alpha) for some #92;alpha#92;inL. The element #92;alpha is called a primitive element for L#47;K. Every finite separable algebraic field extension is simple.
Suppose that L#61;K(#92;alpha#95;1,#92;dots#92;alpha#95;r) is a finite separable extension and K#95;0#92;subseteqK is an infinite subset of K. Then there exists a primitive element #92;alpha of the form #92;textstyle#92;alpha#61;#92;sum#95;#123;i#61;1#125;ʳc#95;i#92;alpha#95;i with c#95;i#92;inK#95;0.

2009, Henning Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer, page 330:

(algebra, field theory) An element that generates a simple extension.

primitiivinen alkio

element that generates a field extension

Furthermore, if the irreducible polynomial has a primitive element α (where α=1) that is a root, then the polynomial is termed a primitive polynomial and corresponds to the polynomial for a maximal length feedback shift register.

1996, J. J. Spilker, Jr. Chapter 3: GPS Signal Structure and Theoretical Performance, Bradford W. Parkinson, James J. Spilker (editors), Global Positioning System: Theory and Applications, Volume 1, AIAA, page 114, Likewise, α=α, etc., namely, the elements are all expressed as powers of α and because α=1, α is termed a primitive element of GF(2ⁿ). […]

2003, Soonhak Kwon, Chang Hoon Kim, Chun Pyu Hong, Efficient Exponentiation for a Class of Finite Fields GF(2ⁿ) Determined by Gauss Periods, Colin D. Walter, Çetin K. Koç, Christof Paar (editors), Cryptographic Hardware and Embedded Systems, CHES 2003: 5th International Workshop, Proceedings, Springer, LNCS 2779, page 228,
Also in the case of a Gauss period of type (n,1), i.e. a type I optimal normal element, we find a primitive element in GF(2ⁿ) which is a sparse polynomial of a type I optimal normal element and we propose a fast exponentiation algorithm which is applicable for both software and hardware purposes.

Here, necessarily, c must be a primitive element of 𝔽_𝕢, since this is the norm of a root of the polynomial.

2008, Stephen D. Cohen, Mateja Preśern, The Hansen-Mullen Primitivity Comjecture: Completion of Proof, James McKee, James Fraser McKee, Chris Smyth (editors, Number Theory and Polynomials, Cambridge University Press, page 89, For q a power of a prime p, let 𝔽_𝕢 be the finite field of order q. Its multiplicative group 𝔽^*_q is cyclic of order q-1 and a generator of 𝔽^*_q is called a primitive element of F_q. More generally, a primitive element γ of F_qⁿ, the unique extension of degree n of 𝔽_𝕢, is the root of a (necessarily monic and automatically irreducible) primitive polynomial f(x)∈ 𝔽_𝕢[x] of degree n. […]

(algebra, field theory, of a finite field) An element that generates the multiplicative group of a given Galois field (finite field).

primitiivinen alkio

element of a finite field that generates its multiplicative group

primitive root of unity

element that generates the multiplicative group of a finite field

But suppose L'#92;inC#95;#92;nu(S#95;0) so that #92;operatorname#123;det#125;(L')#61;#92;eta'#92;pi#92;blacktriangleright 0 for some totally positive unit #92;eta' and so that L' is everywhere locally a primitive''' element of the #92;mathfrako-lattice R#95;#92;nu.

1985, Revista Matemática Iberoamericana, Volume 1, Real Sociedad Matemática Española, page 111:

(algebra, lattice theory, of a lattice) An element that is not a positive integer multiple of another element of the lattice.

primitiivinen alkio

element of a lattice that is not a positive multiple of another element

2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29,
A primitive element of a Hopf algebra is an element h∈H such that
Δh=1⊗h+h⊗1.
It is easily seen that the bracket [x,y]:=xy-yx of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra. For H=U(g) any element of g is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of U(g) coincides with the Lie algebra g.

multiplicative identity
multiplicative identity

(algebra, of a coalgebra over an element g) An element x ∈ C such that μ(x) = x ⊗ g + g ⊗ x, where μ is the comultiplication and g is an element that maps to the multiplicative identity 1 of the base field under the counit (in particular, if C is a bialgebra, g = 1).


primitive polynomial

primitive polynomial (plural primitive polynomials)

We claim that every primitive polynomial can be written as a product of irreducible elements in #92;mathbfD#91;x#93;.[…]By induction on the degree of the primitive polynomials, we conclude that both g(x),h(x) can be written as product of irreducible elements in #92;mathbfD#91;x#93;.

1992, T. T. Moh, Algebra, World Scientific, page 124:

If f(x)#92;in#92;Z#91;x#93;, the ring of polynomials with coefficients in #92;Z, then the content of f, denoted by c(f), is the greatest common divisor of the coefficients of f. The polynomial f(x) is called a primitive polynomial if c(f)#61;1. Since c(fg)#61;c(f)c(g), by Gauss's lemma [Hungerford, 74], the set S of primitive polynomials in #92;Z#91;x#93; is a multiplicatively closed set. Define #92;Lambda#61;#92;Z#91;x#93;#95;S, the localization of #92;Z#91;x#93; at S, a subring of the field of quotients #92;Q(x) of #92;Z#91;x#93;. Elements of #92;Lambda are of the form f(x)#47;g(x) with f(x),g(x)#92;in#92;Z#91;x#93; and g(x) a primitive polynomial.

2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114:

According to the Gauss lemma, the product of primitive polynomials is primitive. Therefore if f(x),g(x) are primitive and f(x)#61;g(x)h(x) with h(x) in F#91;x#93;, then necessarily h(x) is in A#91;x#93; and primitive. […]The irreducible elements of A#91;x#93; are irreducible elements of A and primitive polynomials which are irreducible in F#91;x#93;.

2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1:

greatest common divisor
greatest common divisor

(algebra, ring theory) A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.

monic polynomial

polynomial over an integral domain such that no noninvertible element divides all of its coefficients

Primitive polynomials make the initialization of LFSRs a simpler task since any nonzero state guarantees that all non-zero states will be visited in the maximum length sequence.

2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69:

Definition 7.2 Let f(x) be a monic polynomial of degree n over #92;mathbb#123;F#125;#95;q. If f(x) has a primitive element of #92;mathbb#123;F#125;#95;#123;qⁿ#125; as one of its roots, f(x) is called a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q.
Theorem 7.7 For any positive integer n there always exist primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q. All the n roots of a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q are primitive elements of #92;mathbb#123;F#125;#95;#123;qⁿ#125;. All primitive polynomials of degree n over #92;mathbbF#95;q are irreducible over #92;mathbb#123;F#125;#95;q. The number of primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q is equal to #92;phi(qⁿ-1)#47;n.

2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145:

This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when n = 5, 6, 7 or 8. The HMPC (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed.

2008, Stephen D. Cohen, Mateja Preŝern, “The Hansen-Mullen Primitivity Conjecture: Completion of Proof”, in James McKee, Chris Smyth, editors, Number Theory and Polynomials, Cambridge University Press, page 89:

(algebra, field theory) A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.


polynomial basis

polynomial basis (plural polynomial bases)

2007, Nicos Karcanias, Efstathios Milonidis, 2: Structural Methods for Linear Systems: An Introduction, Matthew C. Turner, Declan G. Bates (editors), Mathematical Methods for Robust and Nonlinear Control: EPSRC Summer School, Springer, Lecture Notes in Control and Information Sciences 367, page 89,
If T(s)=M(s)D(s)⁻¹ is a RCMFD of T(s), then M(s) is a polynomial basis for 𝔛_t. If Q(s) is a greatest right divisor of M(s) then T(s)=◌̅M(s)Q(s)D(s)⁻¹, where ◌̅M(s) is a least degree polynomial basis of 𝔛_t [15].

When dealing with various polynomial bases up to a certain degree and with corresponding moments, any moment (with respect to any basis) can be expressed as a function of moments of the same or fewer orders with respect to an arbitrary basis.[…]As we already saw in Chapter 1, OG^([orthogonal]) moments are, unlike geometric and all other moments, coordinates of f in the polynomial basis in the common sense used in linear algebra.

2009, Jan Flusser, Barbara Zitova, Tomas Suk, Moments and Moment Invariants in Pattern Recognition, Wiley, page 166:

For this purpose, a polynomial basis is used here, which consists of all monomials in the scheduling signals up to a fixed total order. Such a polynomial basis can be interpreted as the polynomial terms, which would appear in a multivariate Taylor approximation of the unknown scheduling function.

2012, A. Kominek et al., “Ident. of Low-Compl. LPV-IO Models for Engine Control”, in Javad Mohammadpour, Carsten W. Scherer, editors, Control of Linear Parameter Varying Systems with Applications, Springer, page 452:

(algebra, ring theory) A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients).

When using polynomial bases, the first stage in computing the product of two elements of #92;mathbbF#95;#123;2ⁿ#125; is the multiplication of two polynomials of degree at most n-1 in #92;mathbbF#95;2#91;x#93;.

1999, I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, page 20:

2010, Vladimir Tujillo-Olaya, Jaime Velasco-Medina, Hardware Architectures for Elliptic Curve Cryptoprocessors Using Polynomial and Gaussian Normal Basis Over GF(2²³³), Marina L. Gravrilova, C. J. Kenneth Tan, Edward David Moreno (editors), Transactions on Computational Science XI: Special Issue on Security in Computing, Part 2, Springer, LNCS 6480, page 79,
In this case, the GF(2ᵐ) multiplication is implemented in hardware using three algorithms for polynomial basis (PB) and three for gaussian normal basis (GNB).

In contrast to the case of normal bases considered in Theorem 5.1.9, the dual basis of a polynomial basis is usually not a polynomial basis.
[…]
5.1.13 Theorem [1265, 1298] Let #92;theta be a root of a monic irreducible polynomial f of degree m over K#61;#92;mathbbF#95;q, and let B#61;#92;#123;1,#92;theta,#92;theta²,#92;dots#92;theta#123;m-1#125;#92;#125; be the corresponding polynomial basis of F#61;#92;mathbbF#95;#123;qᵐ#125; over K. Then the dual basis B#42; of B is a polynomial basis if and only if f is a binomial and m#92;equiv 1#92;#33;#92;#33;#92;#33;#92;#33;#92;pmodp, where q is a power of the prime p.
5.1.14 Corollary There exists a dual pair of polynomial bases of #92;mathbbF#95;#123;qᵐ#125; over #92;mathbbF#95;q if and only if the following three conditions are satisfied:[…].

2013, Gary L. Mullen, Daniel Panario, Handbook of Finite Fields, Taylor & Francis (Chapman & Hall / CRC Press), page 103:

primitive polynomial
primitive polynomial

(algebra, field theory, cryptography, of a finite field) Specifically, a basis, of the form {1, α, ..., αⁿ⁻¹}, of a finite extension F_(qⁿ) of a Galois field F_q, where α is a primitive element of F_(qⁿ) (i.e., a root of a degree-n primitive polynomial over F_q).


perfect field

1984, Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press, Addison-Wesley, page 10,
If K is a perfect field of prime characteristic p, and if n is a nonnegative integer, then the mapping α→α from K to K is an automorphism.

So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.

2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116:

a) K is a perfect field;
b) any irreducible polynomial of K[X] is separable;
c) any element of an algebraic closure of K is separable over K;
d) any algebraic extension of K is separable;
e) for any finite extension K→L, the number of K-homomrphisms from K to an algebraically closed extension of K is equal to [L:K].
Corollary 3.1.8. Any algebraic extension of a perfect field is again a perfect field.

2005, Antoine Chambert-Loir, A Field Guide to Algebra, Springer, page 57, Definition 3.1.7. One says a field K is perfect if any irreducible polynomial in K[X] has as many distinct roots in an algebraic closure as its degree. By the very definition of a perfect field, Theorem 3.1.6 implies that the following properties are equivalent

(algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots.

corps parfait

field such that every irreducible polynomial over it has distinct roots


Frobenius endomorphism

Frobenius endomorphism (plural Frobenius endomorphisms)

Named after German mathematician Ferdinand Georg Frobenius.

Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.

2003, Claudia Miller, “The Frobenius endomorphism and homological dimensions”, in Luchezar L. Avramov, Marc Chardin, Marcel Morales, Claudia Polini, editors, Commutative Algebra: Interactions with Algebraic Geometry: International Conference, American Mathematical Society, page 208:

2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11,
Let k=◌̅ 𝔽_𝕡, and let q be a power of p such that the group G is defined over 𝔽_𝕢. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra 𝒢 and the adjoint action of G on 𝒢 are also defined over 𝔽_𝕢 and we still denote by F:𝒢→𝒢 the Frobenius endomorphism on 𝒢.
[…] Assume that H,X and the action of H over X are all defined over 𝔽_𝕢. Let F:X→X and F:H→H be the corresponding Frobenius endomorphisms.

2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356,
The first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (MEVA 1900) using the curve
E:y²+y=x³.
In this case, the characteristic polynomial of the Frobenius endomorphism denoted by ϕ₂ (cf. Example 4.87 and Section 13.1.8), which sends P_∞ to itself and (x_1,y_1) to (x,y), is
χ_E(T)=T²+2.
Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P∈E( 𝔽_2ᵈ), we have ϕ=-[2]P.

(algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.

commutative algebra; field theory; commutative algebra; field theory

Ferdinand Georg Frobenius
Frobenius endomorphism

endomorphisme de Frobenius

particular endomorphism on a commutative ring with prime characteristic


normal basis

A normal basis is generated by the repeated action of the Frobenius endomorphism on a suitable element β; it is the orbit of β for that endomorphism.

It is a characterising property of normal bases that #92;beta#123;qᵐ#125;#61;#92;beta.

1989, Willi Geiselmann, Dieter Gollmann, Symmetry and Duality in Normal Basis Multiplication, T. Mora (editor), Applied Algebra, Algebraic Algorithms, and Error-correcting Codes: 6th International Conference, Proceedings, Springer, LNCS 357, page 230,
We also combine dual basis and normal basis techniques. The duality of normal bases is shown to be equivalent to the symmetry of the logic array of the serial input / parallel output architectures proposed in this paper.

12.3 In the finite field E#61;#92;mathbbF#95;#123;3³#125;, find: (a) a primitive root of E whose conjugates do not form a normal basis of E#47;#92;mathbbF#95;3; (b) a normal basis that does not consist of primitive roots of E.
For an arbitrary field E with prime field #92;mathbbF#95;p, the extension E#47;#92;mathbbF#95;p does always have at least one normal basis consisting of primitive roots.

2006, Falko Lorenz, translated by Silvio Levy, Algebra: Volume I: Fields and Galois Theory, Springer, page 260:

2015, Sergey Abrahamyan, Melsik Kyureghyan, New recursive construction of normal polynomials over finite fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott (editors, Topics in Finite Fields, American Mathematical Society, page 1,
The set of conjugates of normal element is called normal basis. A monic irreducible polynomial F∈ 𝔽_𝕢[x] is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over 𝔽_𝕢. The minimal polynomial of an element in a normal basis α,α^q,…,α is m(x)=∏ᵢ₌₀ⁿ⁻¹(x-α)∈ 𝔽_𝕢[x] which is irreducible over 𝔽_𝕢. The elements of a normal basis are exactly the roots of some N-polynomial. Hence an N-polynomial is just another way of describing a normal basis.

(algebra, field theory) For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}.

base normale

particular type of basis of a finite field


Galois extension

Galois extension (plural Galois extensions)

Named for its connection with Galois theory and after French mathematician Évariste Galois.

The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.

The fundamental theorem of Galois theory states that there is a one-to-one correspondence between the subfields of a Galois extension and the subgroups of its Galois group.

Corollary If L#58;K is a Galois extension, there exists an irreducible polynomial f in K#91;x#93; such that L#58;K is a splitting field extension for f over K.

1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108:

First, arithmetic obstructions against constructing central extensions of a fixed finite base Galois extension are analyzed with the local-global principle to give some quantitative estimates of them.

1989, Katsuya Miyake, “On central extensions”, in Jean-Marie De Koninck, Claude Levesque, editors, Number Theory, Walter de Gruyter, page 642:

With the help of the results in Section 7.5 it is not hard to describe all Galois extensions.
Proposition 7.6.1. Let E#47;F be a finite field extension. Then (i) E#47;F is a Galois extension if and only if it is normal and separable; (ii) E#47;F is contained in a Galois extension if and only if it is separable.

2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211:

(algebra, Galois theory) An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F.

extension de Galois

algebraic extension that is normal and separable


separable polynomial

separable polynomial (plural separable polynomials)

Over a perfect field, the separable polynomials are precisely the square-free polynomials.

The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.

We know that F(#92;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).

1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:

Proposition 1.4.2 A finite field extension M#47;K is Galois if and only if M is the splitting field over K of a separable polynomial.

2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:

If #92;pi#95;P is a separable polynomial in K#91;t#93;, then the derivative #92;partial#95;t#92;pi#95;P is prime to #92;pi#95;P in K#91;t#93;, and therefore a unit in K#91;t#93;#95;P.[…]In the case when #92;pi#95;P is an inseparable polynomial we may write #92;pi#95;P#61;f(t#123;pʳ#125;) for a suitable rgt;0 and separable polynomial f.

2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:

(algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial).


splitting field

splitting field (plural splitting fields)

Theorem 3.2. If K is a field and f#92;inK#91;x#93; has degree n#92;ge 1, then there exists a splitting field F of f with #91;F#58;K#93;#92;len#33;

1974, Thomas W. Hungerford, Algebra, Springer, page 258:

In the case of the polynomial t²#43;1#92;in#92;R#91;t#93;, the situation is quite clear; its splitting field is #92;C since t²#43;1#61;(t#43;i)(t-i) where i#61;#92;sqrt#123;-1#125;.

2003, Derek J. S. Robinson, An Introduction to Abstract Algebra, Walter de Gruyter, page 130:

Example 4.3. By definition, #92;Q(i) is the splitting field of x²#43;1 over #92;Q, and #92;C is the splitting field for the same polynomial, over #92;R.
Example 4.4. The splitting field F of x⁸-1 over #92;Q is generated by #92;zeta#58;#61;2#123;2#92;pii#47;8#125;; indeed, the roots of x⁸-1 are all the 8-th roots of 1, and all of them are powers of #92;zeta:[…]In fact, #92;zeta is a root of the polynomial x⁴#43;1, which is irreducible over #92;Q; therefore F#61;#92;Q(#92;zeta) is 'already' the splitting field of x⁴#43;1.

2009, Paolo Aluffi, Algebra: Chapter 0, American Mathematical Society, page 430:

(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.

deskonposizio gorputz

(Galois theory, of a polynomial) smallest field containing all roots

The terminology "splitting field of a K-algebra" is motivated by the same terminology regarding a polynomial. A splitting field of a K-algebra A is a field extension K#92;mapstoL such that A#92;otimes#95;KL is split; in the special case A#61;K#91;x#93;#47;f(x) this is the same as a splitting field of the polynomial f(x).

2001, T. Y. Lam, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117,
Ex. 7.6. For a finite-dimensional k-algebra R, let T(R)= operatorname radR+[R,R], where [R,R] denotes the subgroup of R generated by ab-ba for all a,b∈R. Assume that k has characteristic p>0. Show that
T(R)⊆a∈R:aforsomem>1,
with equality if k is a splitting field for R.

Group algebras are defined over the prime field #92;Q or #92;mathbbF#95;p (depending on the characteristic), and by what we have just proved #92;QG and #92;mathbbF#95;pG have splitting fields that are finite degree extensions of the prime field.[…]
Some other basic facts about splitting fields are left to the exercises at the end of this chapter. Thus, if A is a finite-dimensional algebra over a field F that is a splitting field for A and E#92;supsetF is a field extension, it is the case that every simple E#92;otimes#95;FA-module can be written in F (Exercises 4 and 8).

2016, Peter Webb, A Course in Finite Group Representation Theory, Cambridge University Press, page 162:

(algebra, ring theory, of a K-algebra) Given a finite-dimensional K-algebra (algebra over a field), an extension field whose every simple (indecomposable) module is absolutely simple (remains simple after the scalar field has been extended to said extension field).

Every finite dimensional central simple algebra has a splitting field: moreover, if said CSA is a division algebra, then a maximal subfield of it is a splitting field.

1955, Shimshon A. Amitsur, Generic Splitting Fields of Central Simple Algebras, Annals of Mathematics, Volume 62, Number 1, Reprinted in 2001, Avinoam Mann, Amitai Regev, Louis Rowen, David J. Saltman, Lance W. Small (editors), Selected Papers of S. A. Amitsur with Commentary, Part 2, American Mathematical Society, page 199,
The main tool in studying the structure of division algebras, or more generally, of central simple algebras (c.s.as) over a field ℭ are the extensions of ℭ that split the algebras. A field 𝔉⊇ℭ is said to split a c.s.a. 𝔄 if 𝔄⊗𝔉 is a total matrix ring over 𝔉. The present study is devoted to the study of the set of all splitting fields of a given c.s.a. 𝔄.

central simple algebra
central simple algebra

(algebra, ring theory, of a central simple algebra) Given a central simple algebra A over a field K, another field, E, such that the tensor product A⊗E is isomorphic to a matrix ring over E.

1999, P. Shumyatsky, V. Zobina (translators), David Louvish (editor of translation), Ya. G. Berkovich, E. M. Zhmud’, Characters of Finite Groups, Volume 2, American Mathematical Society, page 165,
DEFINITION 2. A field K is called a splitting field of a character χ of a group G if χ∈ operatorname Char_K(G), i.e., χ is afforded by a K-representation of G.
Let T be a representation of G affording the character χ. It follows from Definition 2 that K is a splitting field of χ if and only if T is equivalent to Δ, where Δ is a K-representation of G. In other words, K is a splitting field of a character χ if and only if a representation T affording χ is realized over K. Every character of G has a splitting field (for example, C is a splitting field of any character of G). If K is a splitting field of both characters χ₁,χ₂, then K is a splitting field of χ₁+χ₂, Therefore, in studying splitting fields, we may consider irreducible characters only.
DEFINITION 3. A field K is called a splitting field of a group G if it is a splitting field for every χ∈ operatorname Irr(G).

(algebra, character theory) (of a character χ of a representation of a group G) A field K over which a K-representation of G exists which includes the character χ; (of a group G) a field over which a K-representation of G exists which includes every irreducible character in G.


central simple algebra

central simple algebra (plural central simple algebras)

The complex numbers #92;C form a central simple algebra over themselves, but not over the real numbers #92;R (the centre of #92;C is all of #92;C, not just #92;R). The quaternions #92;mathbbH form a 4-dimensional central simple algebra over #92;R.

The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).

This crossed product E#92;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.

1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 151:

Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra D.

2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:

Let A#95;1,A#95;2 be central simple algebras over a field K. Then A#95;1#92;otimes#95;KA#95;2 can be shown to be a central simple algebra over K. Further, if A is a central simple algebra over a field K, then A#92;otimes#95;KA#92;operatorname#123;op#125;#92;cong#92;operatorname#123;Aut#125;#95;#123;K#92;operatorname#123;-Vect

2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:

associative algebra
associative algebra#English

(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.

centrale enkelvoudige algebra

type of associative algebra over a field


valuation ring

(algebra) An integral domain D such that for every element x of its field of fractions F, at least one of x or x⁻¹ belongs to D.


transcendence degree

transcendence degree (plural transcendence degrees)

2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263,
Let F_1/k and F_2/k denote algebraic function fields of transcendence degree one.

Lemma 7.19 Suppose that L is a field extension^([sic – meaning extension field]) of transcendence degree r over a field K and that L is not separably generated over K. If x#95;1,#92;dots,x#95;n are elements of L such that L#61;K(x#95;1,#92;dots,x#95;n), then for a suitable relabeling of the x#95;i's, the subfield K(x#95;1,#92;dots,x#95;#123;r#43;1#125;) of L is of transcendence degree r and is not separably generated over K.

2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422:

Proposition 2.1.1 Every finite matrix group #92;Gamma#92;subset#92;operatorname#123;GL#125;(#92;Cⁿ) has n algebraically independent invariants, i.e., the ring #92;C#91;x#93;#92;Gamma has transcendence degree n over #92;C.

2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24:

algebraically independent
algebraically independent

(algebra, field theory, of a field extension) Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.

transcendental degree

cardinality of largest algebraically independent subset of a given extension field


algebraically independent

en-adj

algebraically independent (not comparable)

algebraically dependent

antonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”

The singleton set #92;#123;#92;alpha#92;#125; is algebraically independent over K if and only if the element #92;alpha is transcendental over K.

A subset S#92;subsetL is algebraically independent over K if every element of S is transcendental over K and over each of the extension fields over K generated by the remaining elements of S.

1999, David Mumford, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Lecture Notes in Mathematics 1358, 2nd Edition, Expanded, page 40,
If the statement is false, there are n elements x_1,…,x_n in R such that their images ◌̅x_i in R/P are algebraically independent. Let 0 ne p∈P. Then p,x_1,…,x_n cannot be algebraically independent over k, so there is a polynomial P(Y,X_,…,X_n) over k such that P(p,x_,…,x_n)=0.

Setting y#95;i#61;y#95;#123;i,1#125; (where 1 denotes the identity of the semigroup T) we obtain a #92;sigma-algebraically independent over R set #92;#123;y#95;i#92;verti#92;inI#92;#125; such that S#61;R#92;#123;(y#95;i)#95;#123;i#92;inI#125;#92;#125;.

2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251:

If α ne 0,1 is algebraic and β is an algebraic irrational of degree d>2, then
αᵝ,…,α
are algebraically independent.

2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138, Let us begin with the following conjecture of Schneider

(algebra, field theory) (Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K.


stabilizer subgroup

en-noun

stabilizer subgroup (plural stabilizer subgroups)

(algebra) Said of a group acting on a set, and with respect to an element of that set: the subgroup of the given group whose action leaves that element fixed.

(algebra) Said of a group acting on a set, and with respect to a subset of that set: the subgroup of the given group for which the image of the set under the group action is a subset of the original set. Some authors instead define this to be the subgroup of the given group whose action leaves each element of the subset fixed.


orbit-stabilizer theorem

(algebra) A theorem which states that for each element of a given set that a given group acts on, there is a natural bijection between the orbit of that element and the cosets of the stabilizer subgroup with respect to that element.


associative algebra

associative algebra (plural associative algebras)

algebra over a field
algebra over a field

(algebra) An algebra over a ring (or more narrowly, an algebra over a field) whose bilinear product is associative.


algebra over a field

algebra over a field (plural algebras over a field)

(algebra) An algebra over a ring such that the ring is in fact a field. As a module over a field, such an algebra is in particular a vector space.


algebra over a ring

algebra over a ring (plural algebras over rings)

(algebra) A module (over a commutative ring R) equipped with a bilinear binary operation taking values in R.


non-associative algebra

non-associative algebra (plural non-associative algebras)

(algebra) An algebra over a ring (or more narrowly, an algebra over a field) whose bilinear product is not necessarily associative.


free module

(algebra) A module that has a basis. Equivalently, a module consisting of n-tuples of ring elements with no extra identities. (Then the number n is said to be its rank.) Equivalently, a direct sum whose summands are all the same ring. (The quantity of summands may be more than two.)


Ockham algebra

Named after the English Franciscan friar and scholastic philosopher William of Ockham (1287–1347) by Alasdair Urquhart in 1979. The notion was introduced by Joel Berman in 1977.

(algebra) A bounded distributive lattice with a dual endomorphism (where “dual” means that it satisfies De Morgan’s laws).

Alasdair Urquhart
Ockham algebra
William of Ockham


division algebra

division algebra (plural division algebras)

(algebra) A linear algebra such that its non-zero vectors form a multiplicative group.

(algebra) An algebra over a field such that every non-zero element of it has a multiplicative inverse. (It is not required to have a unity element.)


exterior product

exterior product (plural exterior products)

The exterior product of two vectors is a bivector.

The exterior product between two vectors is anti-commutative; therefore the exterior product between a vector and itself is zero.

The exterior product between a multivector of grade k and a multivector of grade n is a multivector of grade k+n, unless k+n is larger than the dimension of the vector space to which the vectors belong (out of which the multivectors are constructed), in which case their product is zero.

The exterior product between a multivector of grade k and a multivector of grade n is commutative if k times n is even and anti-commutative if k times n is odd. (This is related to what is meant when it is said that the exterior product is alternating. It means that a permutation of the factors of a wedge product of vectors changes the sign of the product if and only if the permutation is odd.)

(algebra) A kind of product between vectors and/or multivectors which is associative, linear, and alternating.


lie over

en-verb

lie over (third-person singular simple present lies over, present participle lying over, simple past lay over, past participle lain over)

lies over (present singular third-person)
lying over (participle present)
lay over (past)
lain over (participle past)

Let R be a subring of a ring R′ which is integral over R. If P is a prime ideal of R, then there is a prime ideal P′ of R′ that lies over P. Moreover, if P′ and P″ are prime ideals of R′ that lie over P and if P′ ⊆ P″, then P′ = P″.

1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Academic Press, page 84:

Let R ⊆ S be integral domains such that R is integrally closed and S is integral over R. Suppose that P′ ⊆ P are prime ideals of R. Let Q be a prime ideal of S which lies over P. Then Q contains a prime ideal Q′ of S that lies over P′.

2012, A. Barlotti, E.W. Ellers, P. Plaumann, K. Strambach, Generators and Relations in Groups and Geometries, Springer Science & Business Media, page 369:

(transitive, algebra, ring theory) Of a prime ideal q of a ring R' with subring R, to intersect with R at some other prime ideal p; q lies over p if and only if q∩R=p.


matrix algebra

en-noun

matrix algebra (usually uncountable, plural matrix algebras)

(algebra) Generically, the arithmetic of matrices, involving matrix addition, scalar multiplication of matrices, matrix multiplication, and matrix inversion.

(algebra) A matrix ring over a commutative ring.


circle group

(algebra) The group of all complex numbers of unit modulus, with the group operation being complex multiplication.


double number

split-complex number
split-complex number


polynomial algebra

polynomial algebra (plural polynomial algebras)


Grassmann algebra

Grassmann algebra (plural Grassmann algebras)

Named after Hermann Grassmann (1809–1877), a German polymath.

If a Grassmann algebra’s first grade consists of vectors of dimension n, then the algebra has n grades and 2ⁿ dimensions.

(algebra) A direct sum with an exterior product of multivector spaces which are all based on a same underlying finite-dimensional vector space. The associative algebra of sums and exterior products of scalars, vectors, blades, multivectors, and hybrid (i.e., non-homogeneous) sums of multivectors of different grades.


ring homomorphism

ring homomorphism (plural ring homomorphisms)

(algebra) A function from one ring to another one which preserves the structure of the additive operation, the structure of the multiplicative operation, the zero element (additive identity), and, if there is one, the unity element (multiplicative identity).


ring isomorphism

ring isomorphism (plural ring isomorphisms)

(algebra) A ring homomorphism which is also a bijection.


residue field

(algebra) The quotient ring of a commutative ring divided by one of its maximal ideals; by a certain theorem such a quotient ring must be a field.


Kummer ring

Named after Ernst Kummer (1810–1893), a German mathematician.

(algebra) A ring obtained by adjoining a non-real complex pᵗʰ root of unity (where p is a prime number) to the ring of integers.


simple extension

simple extension (plural simple extensions)

(algebra, field theory) A field extension obtained by adjoining a single element to a given field.


root field

(algebra, Galois theory) splitting field


ring of fractions

ring of fractions (plural rings of fractions)

(algebra) A ring whose elements are fractions whose numerators belong to a given commutative unital ring and whose denominators belong to a multiplicatively closed unital subset D of that given ring. Addition and multiplication of such fractions is defined just as for a field of fractions. A pair of fractions a/b and c/d are deemed equivalent if there is a member x of D such that x(ad-bc)=0.


total ring of fractions

total ring of fractions (plural total rings of fractions)

(algebra) A ring of fractions of a given ring, such that the set of allowed denominators contains all of the non-zero divisors of the given ring.


multiplicative subset

multiplicative subset (plural multiplicative subsets)

If P is a proper prime ideal of a ring and M is the set complement, w.r.t. the ring, of P, then M is a multiplicative subset.

(commutative algebra) A subset of a ring which contains the ring’s unity and which is multiplicatively closed, i.e., it is closed with respect to the ring’s multiplicative operation.


separable extension

separable extension (plural separable extensions)

separable polynomial
separable polynomial

(algebra) A finite (and thereby algebraic) extension of a base field such that every element of the extension is the root of a separable polynomial over the base field.


normal extension

normal extension (plural normal extensions)

(algebra) A finite, algebraic extension of some field such that if it contains any root of any irreducible polynomial over that base field then it contains all of the roots of that polynomial.


fixed field

(algebra, Galois theory) A subfield of a given field which contains all of the fixed points that are common to all of the automorphisms of some subgroup of the automorphism group of that given field.


simple root

(algebra) A root of a polynomial equation which has multiplicity one.


multiple root

(algebra) A root of a polynomial equation which has multiplicity greater than one.


solvable group

(algebra, Galois theory) A group G which is part of a finite chain Gᵢ (0 ≤ i ≤ n) of subgroups, where G₀ is the trivial subgroup and Gₙ = G, such that each intermediate subgroup Gᵢ is a normal subgroup of the next in the chain (i.e., Gᵢ ◁ Gᵢ₊₁) and each quotient Gᵢ₊₁/Gᵢ is a cyclic group.


k-algebra

A k'''-algebra A with ring homomorphism ϕ:k→A is a k-vector space with scalar multiplication (i.e., action of k upon A): λ·a=ϕ(λ)a where λ∈k,a∈A.

(algebra) An algebra over a field; a ring with identity together with an injective ring homomorphism from a field, k, to the ring such that the image of the field is a subset of the center of the ring and such that the image of the field’s unity is the ring’s unity.


direct system

(algebra) A set of algebraic structures (which are part of a concrete category; e.g., modules) and a set of morphisms between them (e.g., module homomorphisms) which all together form a small category which is the image of a covariant functor whose domain is a directed poset.


direct limit

A direct limit has “canonical functions” which map each element of the disjoint union to its equivalence class.

Direct limits in the algebraic sense are models of category-theoretic colimits.

(algebra) A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If i<k,j<k in the indexing poset, then there exist f_ik:A_i→A_k and f_jk:A_j→A_k. If a_i∈A_i,a_j∈A_j such that f_ik(a_i)=f_jk(a_j) then a_i∼a_j. If k = j then f_jj(a_j)=a_j,f_ij(a_i)=a_j.)


projective module

projective module (plural projective modules)

(algebra) A module which is a direct summand of a free module.


Noetherian module

Noetherian module (plural Noetherian modules)

Noetherian modules (plural)
noetherian module (alternative)

(algebra) A module whose every submodule is finitely generated; or equivalently, a module whose submodules satisfy an ascending chain condition (i.e., any ascending chain of submodules levels off after a finite number of steps).


regular element

regular element (plural regular elements)

(algebra) An element of a ring that is not a zero divisor.


Euclid's lemma

Named after ancient Greek mathematician Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Elements.

I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that #92;sqrt 2 is irrational. I said there that if 2 is a factor of a² then a itself must be even. This follows from Euclid's Lemma upon taking p#61;2, the only even prime, and taking b#61;a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing #92;sqrt 2 to be irrational to prove that #92;sqrtp is irrational for any prime p.

1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78:

If a and b are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: 12#92;mid 9#92;cdot 8, but 12#92;nmid 9 and 12#92;nmid 8.

2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179:

In our discussion of Euclid's lemma (Corollary 2.18), we noted that the uniqueness of factorization of integers is a fact that we often take for granted given the way it is introduced in school.

2008, Martin Erickson, Anthony Vazzana, Introduction to Number Theory, Taylor & Francis (Chapman & Hall / CRC Press), page 42:

principal ideal domain
principal ideal domain

(number theory) The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;

The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;
slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
(algebra, by generalisation) the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.


algebraic combinatorics

1982, Trevor Evans, Finite Representations of Two-variable Identities, E. Mendelsohn, Algebraic and Geometric Combinatorics, North-Holland, page 135,
It is part of the folklore of algebraic combinatorics that “most' two-variable groupoid identities have non-trivial models in finite fields, the groupoid operation being represented by a linear function ax+by.

This volume arose through the initiative of Kluwer Academic Publishers in an attempt to introduce some areas of research in algebraic combinatorics which originally appeared in Russian to a wider mathematical community.

1993, I. A. Faradžev, A. A. Ivanov, M. Klin, A. J. Woldar, editors, Investigations in Algebraic Theory of Combinatorial Objects, Kluwer Academic, page vii:

This book is intended primarily as a one-semester undergraduate text for a course in algebraic combinatorics.[…]Algebraic combinatorics is a huge subject, so some selection process was necessary to obtain the present text.

2013, Richard P. Stanley, Algebraic Combinatorics, 2nd edition, Springer, page xi:

(algebra, combinatorics) A branch of mathematics in which techniques from abstract algebra are applied to problems in combinatorics, and vice versa.

combinatoire algébrique

branch of mathematics at the intersection of combinatorics and algebra


Clifford algebra

Clifford algebra (plural Clifford algebras)

Named after William Kingdon Clifford (1845–1879), an English mathematician and philosopher.

(algebra, mathematical physics) A unital associative algebra which generalizes the algebra of quaternions but which is not necessarily a division algebra; it is generated by a set of γᵢ (with i ranging from, say, 1 to n) such that the square of each γᵢ is fixed to be either +1 or −1, depending on each i, and such that any product γᵢγⱼ anticommutes when its factors are distinct (i.e., when i ne j).

Clifford algebra
William Kingdon Clifford


homological algebra

The notion of chain complex is central to homological algebra.

The development of homological algebra was closely intertwined with the emergence of category theory.

Homological algebra for our purposes consists of studying certain abelian groups Hⁿ(G,A) and H#95;n(G,A), known as the nth cohomology and homology groups of the group G with coefficients in the G-module A.

2002, C. R. Leedham-Green, S. McKay, The Structure of Groups of Prime Power Order, Oxford University Press, page 171:

A similar shift of interpretation of CT took place at about the same time in another context, namely homological algebra.

2007, Ralph Krömer, Tool and Object: A History and Philosophy of Category Theory, Springer (Birkhäuser), page 93:

In Section 4.1, we shall introduce some minimal amount of homological algebra needed to launch singular homology.

2014, Anant R. Shastri, Basic Algebraic Topology, Taylor & Francis (CRC Press / Chapman & Hall), page 169:

(algebra) The branch of mathematics dealing with homology and in particular with homological functors and the algebraic structures they entail.


Witt group

Named after German mathematician Ernst Witt (1911–1991), who introduced the concept in 1937.

groupe de Witt

group of equivalence classes found within a field, variety or category

A general method of studying the Witt group of a smooth variety is through the graded group associated to the filtration induced by the filtration of the Witt group of the function field by powers of the fundamental ideal of even rank forms.

1995, Raman Parimala, “Study of quadratic forms - Some connections with geometry”, in S. D. Chatterji, editor, Proceedings of the International Congress of Mathematicians, Zürich m94, Birkhäuser, page 328:

2011, Marco Schlichting, Higher Algebraic K-theory, Guillermo Cortiñas (editor), Topics in Algebraic and Topological K-Theory, Springer, Lecture Notes in Mathematics 2008, page 167,
The second reason for this emphasis is that an analog of the Thomason-Waldhausen Localization Theorem also holds for many other (co-) homology theories besides K-theory, among which Hochschild homology, (negative, periodic, ordinary) cyclic homology [49], topological Hochschild (and cyclic) homology [2], triangular Witt groups [6] and higher Grothendieck–Witt groups [77].

The Witt group of a category with duality is given as the quotient of the isometry classes of symmetric spaces modulo metabolic spaces.[…]For coherent and derived Witt groups, the derived tensor product of complexes gives rise to duality-preserving functions and consequently to pairings in triangular Witt groups, cf. [GN03].

2020, Matthias Wendt, “Oriented Schubert calculus in Chow-Witt rings of Grassmannians”, in Federico Binda, Marc Levine, Manh Toan Nguyen, Oliver Röndigs, editors, Motivic Homotopy Theory and Refined Enumerative Geometry, American Mathematical Society, pages 239–240:

metabolic quadratic space
metabolic quadratic space

(algebra) Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;

Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;
(algebraic geometry, by extension) given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum, modulo the subgroup generated by metabolic spaces
(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.


torsion subgroup

torsion subgroup (plural torsion subgroups)

A standard theorem about finitely generated abelian groups tells us that E(#92;mathbb#123;Q#125;) looks like
#92;qquadE(#92;mathbb#123;Q#125;)#92;cong(#92;mbox#123;FiniteGroup#125;)#92;times#92;underbrace#123;#92;mathbb#123;Z#125;#92;times#92;mathbb#123;Z#125;#92;times...#92;times#92;mathbb#123;Z

2006 July 7, Joseph H. Silverman, An Introduction to the Theory of Elliptic Curves:

(algebra) Subgroup of an abelian group whose generators have finite order.