primitive element

English

Noun

primitive element (plural primitive elements)

(algebra, field theory) An element that generates a simple extension.
- 2009, Henning Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer, page 330:
  An algebraic extension $L / K$ is called simple if $L = K (α)$ for some $α \in L$ . The element $α$ is called a primitive element for $L / K$ . Every finite separable algebraic field extension is simple.
  Suppose that $L = K (α_{1}, \dots α_{r})$ is a finite separable extension and $K_{0} \subseteq K$ is an infinite subset of $K$ . Then there exists a primitive element $α$ of the form $α = \sum_{i = 1}^{r} c_{i} α_{i}$ with $c_{i} \in K_{0}$ .
(algebra, field theory, of a finite field) 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, AIAA, 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 $G F (2^{n})$ . […]
  
  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, 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 $G F (2^{n})$ 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, 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 $F_{q^{n}}$ , 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 $n$ .
  
  […]
  
  Here, necessarily, $c$ must be a primitive element of $𝔽_{q}$ , since this is the norm of a root of the polynomial.
(number theory) 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, W. Wesley Peterson; E. J. Weldon, Jr., Error-correcting Codes, 2nd edition, The MIT Press, page 457:
  Let $A$ be a prime number for which $2$ is a primitive element. Then $2^{A - 1} - 1$ is divisible by $A$ .
(algebra, lattice theory, of a lattice) An element that is not a positive integer multiple of another element of the lattice.
- 1985, Revista Matemática Iberoamericana, Volume 1, Real Sociedad Matemática Española, page 111:
  But suppose $L^{'} \in C_{ν} (S_{0})$ so that $\det (L^{'}) = η^{'} π ▸ 0$ for some totally positive unit $η^{'}$ and so that $L^{'}$ is everywhere locally a primitive element of the $𝔬$ -lattice $R_{ν}$ .
(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).
- 2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, 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 $𝔤$ .
(group theory, of a free group) An element of a free generating set of a given free group.
- 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:
  In this paper we apply regression models and other pattern recognition techniques to the task of classifying primitive elements of a free group.

Usage notes

(element that generates a field extension): A field extension that is generated by some (single) element is called a simple extension.
(generator of a finite field): A polynomial whose roots are primitive elements (and especially the minimal polynomial of some primitive element) is called a primitive polynomial.

Synonyms

(number theory): primitive root
(element that generates a field extension): generating element
(element that generates the multiplicative group of a finite field): primitive root of unity

Related terms

Translations

element that generates a field extension

Finnish: primitiivinen alkio
Polish: element pierwotny m

element of a finite field that generates its multiplicative group

Finnish: primitiivinen alkio
Polish: element pierwotny m

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

Finnish: primitiivinen alkio
Polish: element pierwotny m

element of a coalgebra satisfying a particular condition

Finnish: primitiivinen alkio
Polish: element pierwotny m