primitive element
Word Rarity
Rare
Frequency: Zipf frequency 3.71 in modern English
25
Scrabble Score
16
Letters
4
Syllables
9
Unique Chars
Meaning
noun
An element that generates a simple extension.
noun
An element that generates the multiplicative group of a given Galois field (finite field).
noun
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.
noun
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).
noun
An element of a free generating set of a given free group.
Example Sentences
- "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."