primitive element

Noun

primitive element (plural primitive elements)

1. (field theory) An element that generates a simple extension.
   • An algebraic extension L#47;K is called simple if L#61;K(#92;alpha) for some #92;alpha#92;inL. The element #92;alpha is called a primitive element for L#47;K. Every finite separable algebraic field extension is simple.
   • Suppose that L#61;K(#92;alpha#95;1,#92;dots#92;alpha#95;r) is a finite separable extension and K#95;0#92;subseteqK is an infinite subset of K. Then there exists a primitive element #92;alpha of the form #92;textstyle#92;alpha#61;#92;sum#95;#123;i#61;1#125;ʳc#95;i#92;alpha#95;i with...2009, Henning Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer, page 330

   Synonyms:

   generating element

2. (field theory) An element that generates the multiplicative group of a given Galois field (finite field).
   • 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.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...
   • 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 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.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)...

   Synonyms:

   primitive root of unity

3. 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.
   • Let A be a prime number for which 2 is a primitive element. Then 2#123;A-1#125;-1 is divisible by A.1972, W. Wesley Peterson; E. J. Weldon, Jr., Error-correcting Codes, 2nd edition, The MIT Press, page 457
4. (lattice theory) An element that is not a positive integer multiple of another element of the lattice.
   • But suppose L'#92;inC#95;#92;nu(S#95;0) so that #92;operatorname#123;det#125;(L')#61;#92;eta'#92;pi#92;blacktriangleright 0 for some totally positive unit #92;eta' and so that L' is everywhere locally a primitive element of the #92;mathfrako-lattice R#95;#92;nu.1985, Revista Matemática Iberoamericana, Volume 1, Real Sociedad Matemática Española, page 111
5. 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).
   • A primitive element of a Hopf algebra is an element h∈H such that
     Δh=1⊗h+h⊗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(g) any element of g is primitive and in fact using the...2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29
6. An element of a free generating set of a given free group.
   • In this paper we apply regression models and other pattern recognition techniques to the task of classifying primitive elements of a 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...