primitive element

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Primitive element

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

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

     Furthermore, if the irreducible polynomial has a primitive element ${\displaystyle \alpha }$ (where ${\displaystyle \alpha ^{2^{n}-1}=1}$) 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 ${\displaystyle (n,1)}$, i.e. a type I optimal normal element, we find a primitive element in ${\displaystyle \mathrm {GF} (2^{n})}$ 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 ${\displaystyle q}$ a power of a prime ${\displaystyle p}$, let ${\displaystyle \mathbb {F} _{q}}$ be the finite field of order ${\displaystyle q}$. Its multiplicative group ${\displaystyle \mathbb {F} _{q}^{*}}$ is cyclic of order ${\displaystyle q-1}$ and a generator of ${\displaystyle \mathbb {F} _{q}^{*}}$ is called a primitive element of ${\displaystyle F_{q}}$. More generally, a primitive element ${\displaystyle \gamma }$ of ${\displaystyle F_{q^{n}}}$, the unique extension of degree ${\displaystyle n}$ of ${\displaystyle \mathbb {F} _{q}}$, is the root of a (necessarily monic and automatically irreducible) primitive polynomial ${\displaystyle f(x)\in \mathbb {F} _{q}}$ of degree ${\displaystyle n}$.

     Here, necessarily, ${\displaystyle c}$ must be a primitive element of ${\displaystyle \mathbb {F} _{q}}$, 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 ${\displaystyle A}$ be a prime number for which ${\displaystyle 2}$ is a primitive element. Then ${\displaystyle 2^{A-1}-1}$ is divisible by ${\displaystyle A}$.

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

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 ${\displaystyle h\in H}$ such that

     ${\displaystyle \Delta h=1\otimes h+h\otimes 1}$.

     It is easily seen that the bracket ${\displaystyle :=xy-yx}$ of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra. For ${\displaystyle H=U({\mathfrak {g}})}$ any element of ${\displaystyle {\mathfrak {g}}}$ is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of ${\displaystyle U({\mathfrak {g}})}$ coincides with the Lie algebra ${\displaystyle {\mathfrak {g}}}$.

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

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