Primitive element

English

Template:Wikipedia

Noun

Template:En-noun

Template:Lb An element that generates a simple extension.
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Template:Lb An element that generates the multiplicative group of a given Galois field (finite field).
- 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, Template:W, page 114,
  Likewise, $α^{2^{n}} = α$ , etc., namely, the elements are all expressed as powers of $α$ and because $α^{2^{n} - 1} = 1$ , $α$ is termed a primitive element of $G F (2^{n})$ .Template:...
  
  Furthermore, if the irreducible polynomial has a primitive element $α$ (where $α^{2^{n} - 1} = 1$ ) that is a root, then the polynomial is termed a primitive polynomial and corresponds to the polynomial for a maximal length feedback shift register.
- 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, Template:W 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 $G F (2^{n})$ 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.
- 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, Template:W, 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 $F_{q^{n}}$ , the unique extension of degree $n$ of $𝔽_{q}$ , is the root of a (necessarily monic and automatically irreducible) primitive polynomial $f (x) \in 𝔽_{q} [x]$ of degree $n$ .
  
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  Here, necessarily, $c$ must be a primitive element of $𝔽_{q}$ , since this is the norm of a root of the polynomial.
Template:Lb Given a modulus n, a number g such that every number coprime to n is congruent (modulo n) to some power of g; equivalently, a generator of the multiplicative field of integers modulo n.
- 1972, Template:W, E. J. Weldon, Jr., Error-correcting Codes, Template:W, 2nd Edition, page 457,
  Let $A$ be a prime number for which $2$ is a primitive element. Then $2^{A - 1} - 1$ is divisible by $A$ .
Template:Lb An element that is not a positive integer multiple of another element of the lattice.
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Template:Lb 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).
- 2009, Masoud Khalkhali, Basic Noncommutative Geometry, Template:W, page 29,
  A primitive element of a Hopf algebra is an element $h \in H$ such that
  $Δ h = 1 \otimes h + h \otimes 1$ .
  
  It is easily seen that the bracket $[x, y] : = x y - y x$ 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 $𝔤$ .
Template:Lb An element of a free generating set of a given free group.
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Usage notes

Template:Sense A field extension that is generated by some (single) element is called a Template:M.
Template:Sense A polynomial whose roots are primitive elements (and especially the minimal polynomial of some primitive element) is called a Template:M.