See primitive element on Wiktionary
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"text": "An algebraic extension L#x2F;K is called simple if L#x3D;K(#x5C;alpha) for some #x5C;alpha#x5C;inL. The element #x5C;alpha is called a primitive element for L#x2F;K. Every finite separable algebraic field extension is simple.\nSuppose that L#x3D;K(#x5C;alpha#x5F;1,#x5C;dots#x5C;alpha#x5F;r) is a finite separable extension and K#x5F;0#x5C;subseteqK is an infinite subset of K. Then there exists a primitive element #x5C;alpha of the form #x5C;textstyle#x5C;alpha#x3D;#x5C;sum#x5F;#x7B;i#x3D;1#x7D;ʳc#x5F;i#x5C;alpha#x5F;i with c#x5F;i#x5C;inK#x5F;0.",
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"ref": "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). […]",
"text": "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."
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"text": "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,\nAlso 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."
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"ref": "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. […]",
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"text": "2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29,\nA primitive element of a Hopf algebra is an element h∈H such that\nΔh=1⊗h+h⊗1.\nIt 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."
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"ref": "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). […]",
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"glosses": [
"An element that is not a positive integer multiple of another element of the lattice."
],
"links": [
[
"algebra",
"algebra"
],
[
"element",
"element"
],
[
"lattice",
"lattice"
]
],
"qualifier": "lattice theory",
"raw_glosses": [
"(algebra, lattice theory, of a lattice) An element that is not a positive integer multiple of another element of the lattice."
],
"raw_tags": [
"of a lattice"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra"
],
"examples": [
{
"text": "2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29,\nA primitive element of a Hopf algebra is an element h∈H such that\nΔh=1⊗h+h⊗1.\nIt 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."
}
],
"glosses": [
"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)."
],
"links": [
[
"algebra",
"algebra"
],
[
"element",
"element"
],
[
"comultiplication",
"comultiplication"
],
[
"multiplicative identity",
"multiplicative identity"
],
[
"counit",
"counit"
],
[
"bialgebra",
"bialgebra"
]
],
"raw_glosses": [
"(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)."
],
"raw_tags": [
"of a coalgebra over an element g"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"English terms with quotations",
"en:Group theory"
],
"examples": [
{
"ref": "2004, Dmitry Y. Bormotov, “Experimenting with Primitive Elements in F₂”, in Alexandre Borovik, Alexei G. Myasnikov, editors, Computational and Experimental Group Theory: AMS-ASL Joint Special Session, American Mathematical Society, page 215:",
"text": "In this paper we apply regression models and other pattern recognition techniques to the task of classifying primitive elements of a free group.",
"type": "quote"
}
],
"glosses": [
"An element of a free generating set of a given free group."
],
"links": [
[
"group theory",
"group theory"
],
[
"free group",
"free group"
],
[
"free generating set",
"free generating set"
]
],
"raw_glosses": [
"(group theory, of a free group) An element of a free generating set of a given free group."
],
"raw_tags": [
"of a free group"
],
"topics": [
"group-theory",
"mathematics",
"sciences"
]
}
],
"synonyms": [
{
"topics": [
"number-theory",
"mathematics",
"sciences"
],
"word": "primitive root"
},
{
"sense": "element that generates a field extension",
"word": "generating element"
},
{
"sense": "element that generates the multiplicative group of a finite field",
"word": "primitive root of unity"
}
],
"translations": [
{
"code": "fi",
"lang": "Finnish",
"sense": "element that generates a field extension",
"word": "primitiivinen alkio"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a finite field that generates its multiplicative group",
"word": "primitiivinen alkio"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a lattice that is not a positive multiple of another element",
"word": "primitiivinen alkio"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a coalgebra satisfying a particular condition",
"word": "primitiivinen alkio"
}
],
"wikipedia": [
"Primitive element"
],
"word": "primitive element"
}
Download raw JSONL data for primitive element meaning in All languages combined (9.4kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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