Meaning of primitive element | Babel Free

Definitions

1. An element that generates a simple extension.
2. An element that generates the multiplicative group of a given Galois field (finite field).
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.
4. An element that is not a positive integer multiple of another element of the lattice.
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).
6. An element of a free generating set of a given free group.

Examples

“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.”

“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⟳.”

“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.”

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

“Let⟳ A be a prime number⟳ for which 2 is a primitive element. Then 2#123;A-1#125;-1 is divisible by A.”

“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.”

“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.”

“In this paper we apply⟳ regression models and other pattern recognition techniques to the task of classifying primitive elements of a free group.”

CEFR level