primitive element

plural primitive elements

1. field theoryalgebraAn element that generates a simple extension.

   1 more example

   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 c#95;i#92;inK#95;0.

   2009, Henning Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer, page 330

   Synonyms: generating element

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

   Here, necessarily, c must be a primitive element of 𝔽_𝕢, since this is the norm of a root of the polynomial.

   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_qⁿ, the unique extension of degree n of 𝔽_𝕢, is the root of a (necessarily monic and automatically irreducible) primitive polynomial f(x)∈ 𝔽_𝕢[x] of degree n. […]

   2 more examples

   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, AIAA, page 114, Likewise, α=α, etc., namely, the elements are all expressed as powers of α and because α=1, α is termed a primitive element of GF(2ⁿ). […]

   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 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.

   Synonyms: primitive root of unity

3. mathematicsGiven 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 theoryalgebraAn 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. algebraAn 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).

   1 more example

   2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29, 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 Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of U(g) coincides with the Lie algebra g.

6. group theoryAn 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 Session, American Mathematical Society, page 215