primitive element

English

Noun

primitive element (plural primitive elements)${\displaystyle }$

1. (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 $L=K(\alpha )$ for some $\alpha \in L$. The element $\alpha$ is called a primitive element for $L/K$. Every finite separable algebraic field extension is simple.
     Suppose that $L=K(\alpha _{1},\dots \alpha _{r})$ is a finite separable extension and $K_{0}\subseteq K$ is an infinite subset of $K$. Then there exists a primitive element $\alpha$ of the form $\textstyle \alpha =\sum _{i=1}^{r}c_{i}\alpha _{i}$ with $c_{i}\in K_{0}$.

2. (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, $\alpha ^{2^{n}}=\alpha$, etc., namely, the elements are all expressed as powers of $\alpha$ and because $\alpha ^{2^{n}-1}=1$, $\alpha$ is termed a primitive element of $\mathrm {GF} (2^{n})$. […]

     Furthermore, if the irreducible polynomial has a primitive element $\alpha$ (where $\alpha ^{2^{n}-1}=1$) 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 $\mathrm {GF} (2^{n})$ 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 $\mathbb {F} _{q}$ be the finite field of order $q$. Its multiplicative group $\mathbb {F} _{q}^{*}$ is cyclic of order $q-1$ and a generator of $\mathbb {F} _{q}^{*}$ is called a primitive element of $F_{q}$. More generally, a primitive element $\gamma$ of $F_{q^{n}}$, the unique extension of degree $n$ of $\mathbb {F} _{q}$, is the root of a (necessarily monic and automatically irreducible) primitive polynomial $f(x)\in \mathbb {F} _{q}[x]$ of degree $n$.

     […]

     Here, necessarily, $c$ must be a primitive element of $\mathbb {F} _{q}$, since this is the norm of a root of the polynomial.

3. (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, The MIT Press, 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 $A$.

4. (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 $L'\in C_{\nu }(S_{0})$ so that $\operatorname {det} (L')=\eta '\pi \blacktriangleright 0$ for some totally positive unit $\eta '$ and so that $L'$ is everywhere locally a primitive element of the ${\mathfrak {o}}$-lattice $R_{\nu }$.

5. (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

     $\Delta h=1\otimes h+h\otimes 1$.

     It 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 $H=U({\mathfrak {g}})$ any element of ${\mathfrak {g}}$ is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of $U({\mathfrak {g}})$ coincides with the Lie algebra ${\mathfrak {g}}$.

6. (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

• primitive polynomial
• primitive root

Translations

element that generates a field extension

• Finnish: primitiivinen alkio

element of a finite field that generates its multiplicative group

• Finnish: primitiivinen alkio

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

• Finnish: primitiivinen alkio

element of a coalgebra satisfying a particular condition

• Finnish: primitiivinen alkio

Further reading

• Primitive element (finite field) on Wikipedia.Wikipedia
• Primitive element (co-algebra) on Wikipedia.Wikipedia
• Simple extension on Wikipedia.Wikipedia
• Primitive root (disambiguation) on Wikipedia.Wikipedia
• Galois field structure on Encyclopedia of Mathematics
• Primitive element in a co-algebra on Encyclopedia of Mathematics