See primitive element in All languages combined, or 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."
},
{
"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. […]",
"text": "Here, necessarily, c must be a primitive element of 𝔽_𝕢, since this is the norm of a root of the polynomial."
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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. […]",
"text": "Here, necessarily, c must be a primitive element of 𝔽_𝕢, since this is the norm of a root of the polynomial."
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"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"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 using wiktextract (f4fd8c9 and c9440ce). 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.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.