Meaning of primitive polynomial | Babel Free

Definitions

1. A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.
2. A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.

Examples

“We claim that every primitive polynomial can be written as a product of irreducible elements in #92;mathbfD#91;x#93;.[…]By induction on the degree of the primitive polynomials, we conclude that both g(x),h(x) can be written as product of irreducible elements in #92;mathbfD#91;x#93;.”

“If f(x)#92;in#92;Z#91;x#93;, the ring of polynomials with coefficients in #92;Z, then the content of f, denoted by c(f), is the greatest common divisor of the coefficients of f. The polynomial f(x) is called a primitive polynomial if c(f)#61;1. Since c(fg)#61;c(f)c(g), by Gauss's lemma [Hungerford, 74], the set S of primitive polynomials in #92;Z#91;x#93; is a multiplicatively closed set. Define #92;Lambda#61;#92;Z#91;x#93;#95;S, the localization of #92;Z#91;x#93; at S, a subring of the field of quotients #92;Q(x) of #92;Z#91;x#93;. Elements of #92;Lambda are of the form f(x)#47;g(x) with f(x),g(x)#92;in#92;Z#91;x#93; and g(x) a primitive polynomial.”

“According to the Gauss lemma, the product of primitive polynomials is primitive. Therefore if f(x),g(x) are primitive and f(x)#61;g(x)h(x) with h(x) in F#91;x#93;, then necessarily h(x) is in A#91;x#93; and primitive. […]The irreducible elements of A#91;x#93; are irreducible elements of A and primitive polynomials which are irreducible in F#91;x#93;.”

“Primitive polynomials make the initialization of LFSRs a simpler task since any nonzero state guarantees that all non-zero states will be visited in the maximum length sequence.”

“Definition 7.2 Let f(x) be a monic polynomial of degree n over #92;mathbb#123;F#125;#95;q. If f(x) has a primitive element of #92;mathbb#123;F#125;#95;#123;qⁿ#125; as one of its roots, f(x) is called a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q. Theorem 7.7 For any positive integer n there always exist primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q. All the n roots of a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q are primitive elements of #92;mathbb#123;F#125;#95;#123;qⁿ#125;. All primitive polynomials of degree n over #92;mathbbF#95;q are irreducible over #92;mathbb#123;F#125;#95;q. The number of primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q is equal to #92;phi(qⁿ-1)#47;n.”

“This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when n = 5, 6, 7 or 8. The HMPC (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed.”

CEFR level