rational root theorem

The rational root theorem states that if the rational number p#47;q is a root of the polynomial equation a#95;nxⁿ#43;a#95;#123;n-1#125;x#123;n-1#125;#43;#92;cdots#43;a#95;0#61;0, with a#95;0,#92;ldotsa#95;n#92;in#92;mathbb#123;Z#125;, then p#92;verta#95;0 and q#92;verta#95;n.

is irreducible over Q [x].

2015, Marlow Anderson, Todd Feil, A First Course in Abstract Algebra, Taylor & Francis (CRC Press), 3rd Edition, page 50, Use the Rational Root Theorem 5.6 to argue that x³+x+7

We demonstrate how to use the rational root theorem to prove the irrationality of many numbers.

2017, Martin H. Weissman, An Illustrated Theory of Numbers, American Mathematical Society, page 84:

The rational root theorem has you create a list of possible rational roots using the lead coefficient and constant in the polynomial equation.

2018, Mary Jane Sterling, Pre-Calculus For Dummies, Wiley, page 85:

(algebra) A theorem which states a constraint on rational solutions of a polynomial equation with integer coefficients.


eigensection

(algebra) A homogenous section whose associated Eulerian family diagonalizes a given element of a Lie algebra.


goat grazing problem

en-noun

goat grazing problem (plural goat grazing problems)

The same problem generalized to any number of dimensions: given a unit n-sphere (n-dimensional hypersphere) S, and a point P on the edge of S, what is r such that exactly one-half of the space (content) within S, is within distance r of P?


goat problem

goat grazing problem
goat grazing problem#English


Bring radical

Named after Swedish mathematician Erland Samuel Bring (1736–1798).

named-after

(algebra) Synonym of ultraradical (“root of the polynomial x⁵ + x + a, where a is a complex number”).


Cartesian equation

en-noun

Cartesian equation (plural Cartesian equations)

(geometry, algebra) An equation that describes a line or plane in terms of the relationship between the coordinates on the axes involved.


Auslander-Reiten theory

Introduced by Maurice Auslander and Idun Reiten (1975).

representation theory
representation theory

Auslander-Reiten sequence
Auslander-Reiten sequence

almost split sequence
almost split sequence

Auslander-Reiten quiver
Auslander-Reiten quiver

(algebra) The study of the representation theory of Artinian rings using techniques such as Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers.


almost split sequence

almost split sequence (plural almost split sequences)

Auslander-Reiten sequence
Auslander-Reiten sequence#English

(algebra) Synonym of Auslander-Reiten sequence.


Abel's theorem

(mathematics, algebra) A theorem that relates a limit of a power series to the sum of its coefficients.


Cohen-Macaulay

Named for Irvin Cohen and Francis Sowerby Macaulay, who proved unmixedness results for specific classes of rings, which Cohen-Macaulay rings generalize.

(commutative algebra, of a finite module over a noetherian local ring) Such that its depth is equal to its Krull dimension.


Krull dimension

Named for Wolfgang Krull, for his work on the height of finitely generated ideals of noetherian rings. The term dimension is from the correspondence between the geometric dimension of an affine variety and the Krull dimension of its coordinate ring.

(commutative algebra, of a commutative ring) The supremum of the lengths of all chains of prime ideals.


codepth

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(commutative algebra, of a finitely generated module over a local ring) The Krull dimension minus the depth.


integrally closed

(commutative algebra, of an integral domain) Equal to its own integral closure in its field of fractions; equivalently, containing all integral elements of its field of fractions.


uniformizing parameter

uniformizing parameter (plural uniformizing parameters)

(commutative algebra) An element of a discrete valuation ring which generates the unique maximal ideal.


finitely generated

From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.

(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:

(most generally, category theory, of an object X in a locally small category 𝒞 that admits filtered colimits of monomorphism) Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.

(less generally, but still covering most cases outside pure category theory, of an object X in a concrete category) Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set.

less generally; but still covering most cases outside pure category theory; less generally; but still covering most cases outside pure category theory

(abstract algebra, of a group-like structure: a group, module, ring, monoid, etc.) Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)

(ring theory, of a (left) ideal in a ring R) Finitely generated as a (left) module over R.

(of a space) Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).


quasi-isomorphism

(homological algebra, of a (co)chain map) Which induces isomorphisms of the (co)homology groups.


formal sum

(mathematics) An expression which looks like a sum but where the symbol + has a purely symbolic role, in the sense that the summands cannot be meaningfully combined.

(algebra) An element of a free abelian group or vector space regarded as a sum of basis elements.


associated prime

associated prime (plural associated primes)

(algebra, of a module M over a ring R) A ideal of R which is the annihilator of some prime submodule of M.

(commutative algebra) If the ring R is commutative, equivalently: A prime ideal of R which is the annihilator of some element of M.


prime subring

The image of the homomorphism ℤ→ R is called the prime subring of R. Clearly ch(R) = 0 ⟷ the prime subring of R is isomorphic to ℤ.

2006, Shreeram Shankar Abhyankar, The Algebraic Structure of Crossed Products, World Scientific, page 21:

The image of the morphism f_A is the prime subring of A. The prime subring is either isomorphic to the ring ℤ or isomorphic to the ring ℤ/nℤ for some n > 0.

2013, Michel Broué, Some Topics in Algebra: An Advanced Undergraduate Course at PKU, Springer Science & Business Media, page 10:

(algebra, ring theory, of a (unital) ring R) The image of the natural homomorphism from ℤ into R, which is the unique homomorphism mapping 1_ℤ to 1_R. Equivalently, the intersection of all subrings of R.


Bézout domain

So-called because such domains satisfy Bézout's identity, in turn named after Étienne Bézout.

(algebra, ring theory) An integral domain in which the sum of two principal ideals is also a principal ideal.


lie above

en-verb

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

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

If p₁, p ∈ Spec(A) with p₁ ⊂ p, and if q₁ ∈ Spec(B) lies above p₁, then there exists q ∈ Spec(B) which lies above p and such that q₁ ⊂ q.

2007, Meinolf Geck, Group Representation Theory, EPFL Press, page 199:

(algebra, ring theory) Synonym of lie over.


discrete valuation ring

en-noun

discrete valuation ring (plural discrete valuation rings)

principal ideal domain
principal ideal domain

(algebra, ring theory) A local principal ideal domain whose unique maximal ideal is not the zero ideal.


zero ideal

(algebra, ring theory, of a ring) The ideal containing only the element zero.


comaximal

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Two proper ideals I and J of R are said to be comaximal (or I is comaximal to J) if I + J = R.

2001, László Fuchs, Luigi Salce, Modules over Non-Noetherian Domains, American Mathematical Society, page 3:

(algebra, ring theory, of two ideals) Synonym of coprime.


-nomial

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Derived from Ancient Greek νομός (nomós, “portion, part”), compare binomial.

der+

(algebra, mathematics) Denotes an expression with a specified number of terms.


inter-universal Teichmüller theory

en-noun

inter-universal Teichmüller theory (uncountable)

Coined by Shinichi Mochizuki, from inter-universal and Teichmüller theory.

The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve – which we refer to as "inter-universal Teichmüller theory" – by applying the theory of semi-graphs of anabelioids, Frobenioids, the étale theta function, and log-shells developed in earlier papers by the author.

2021, Shinichi Mochizuki, “Inter-universal Teichmüller theory I: Construction of Hodge theaters”, in Publications of the Research Institute for Mathematical Sciences, volume 57, number 1, →ISSN, pages 3-207:

(mathematics) an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve


quaternion algebra

quaternion algebra (plural quaternion algebras)

(mathematics, algebra) A particular kind of algebra, generalizing the classical quaternions: a four-dimensional central simple algebra over a field.


equational variety

equational variety (plural equational varieties)

(algebra) Synonym of variety (“an equational class in universal algebra”).


variety of algebras

variety of algebras (plural varieties of algebras)


equational class

equational class (plural equational classes)


primordial element

primordial element (plural primordial elements)

(mathematics, algebra) A particular kind of a vector in a vector space.


Pythagorean field

Pythagorean field (plural Pythagorean fields)

(mathematics, algebra) A field in which every sum of two squares is a square.


descending chain condition

descending chain condition (countable and uncountable, plural descending chain conditions)

partially ordered set
partially ordered set

(algebra) A property of a partially ordered set that states that any decreasing sequence of elements must eventually become constant.


ascending chain condition

ascending chain condition (countable and uncountable, plural ascending chain conditions)

(algebra) A property of a partially ordered set that states that any increasing sequence of elements must eventually become constant.


finitely presented

(algebra, of a module M over a ring R) Isomorphic to the kernel of a morphism of free modules; such that there exists an exact sequence Rᵐ→Rⁿ→M→0 for some integers m,n.


closure operator

closure operator (plural closure operators)

1997 [Kluwer Academic], R. Friĉ, History of Sequential Convergence Spaces, C. E. Aull, R. Lowen (editors), Handbook of the History of General Topology, Volume 1, Springer, page 346,
Let u be a closure operator on X satisfying the first three closure axioms: u(U∪B)=uU∪uB,A⊂u(A),u empty = empty .

(mathematics, topology, algebra, "on" a set S) A function which maps any subset X of S onto another such subset (or possibly X itself) called the closure of X (thus providing a mathematical means to define closure).