primitive element

Noun
$$$$


 * 1)  An element that generates a simple extension.
 * 2)  An element that generates the multiplicative group of a given Galois field (finite field).
 * 3) * 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,, page 114,
 * Likewise, $$\alpha^{2^n}=\alpha$$, etc., namely, the elements are all expressed as powers of $$\alpha$$ and because $$\alpha^{2^n-1}=1$$, $$\alpha$$ is termed a primitive element of $$\mathrm{GF}(2^n)$$.
 * Furthermore, if the irreducible polynomial has a primitive element $$\alpha$$ (where $$\alpha^{2^n-1} = 1$$) that is a root, then the polynomial is termed a primitive polynomial and corresponds to the polynomial for a maximal length feedback shift register.
 * 1) * 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, 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 $$\mathrm{GF}(2^n)$$ 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.
 * 1) * 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,, page 89,
 * For $$q$$ a power of a prime $$p$$, let $$\mathbb F_q$$ be the finite field of order $$q$$. Its multiplicative group $$\mathbb F^*_q$$ is cyclic of order $$q-1$$ and a generator of $$\mathbb F^*_q$$ is called a primitive element of $$F_q$$. More generally, a primitive element $$\gamma$$ of $$F_{q^n}$$, the unique extension of degree $$n$$ of $$\mathbb F_q$$, is the root of a (necessarily monic and automatically irreducible) primitive polynomial $$f(x)\in\mathbb F_q[x]$$ of degree $$n$$.
 * Here, necessarily, $$c$$ must be a primitive element of $$\mathbb F_q$$, since this is the norm of a root of the polynomial.
 * 1)  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.
 * 2) * 1972,, E. J. Weldon, Jr., Error-correcting Codes, , 2nd Edition, page 457,
 * Let $$A$$ be a prime number for which $$2$$ is a primitive element. Then $$2^{A-1}-1$$ is divisible by $$A$$.
 * 1)  An element that is not a positive integer multiple of another element of the lattice.
 * 2)  An element x ∈ C such that μ(x) = x &#X2297; g + g &#X2297; 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).
 * 3) * 2009, Masoud Khalkhali, Basic Noncommutative Geometry,, page 29,
 * A primitive element of a Hopf algebra is an element $$h\in H$$ such that
 * $$\Delta h = 1\otimes h + h\otimes 1$$.
 * 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 $$H = U(\mathfrak g)$$ any element of $$\mathfrak g$$ is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of $$U(\mathfrak g)$$ coincides with the Lie algebra $$\mathfrak g$$.
 * 1)  An element of a free generating set of a given free group.
 * $$\Delta h = 1\otimes h + h\otimes 1$$.
 * 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 $$H = U(\mathfrak g)$$ any element of $$\mathfrak g$$ is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of $$U(\mathfrak g)$$ coincides with the Lie algebra $$\mathfrak g$$.
 * 1)  An element of a free generating set of a given free group.

Usage notes

 * A field extension that is generated by some (single) element is called a.
 * A polynomial whose roots are primitive elements (and especially the minimal polynomial of some primitive element) is called a.

Translations

 * Finnish: primitiivinen alkio


 * Finnish: primitiivinen alkio


 * Finnish: primitiivinen alkio


 * Finnish: primitiivinen alkio