primitive polynomial

plural primitive polynomials

1. ring theoryalgebraA 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.

   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;.

   1992, T. T. Moh, Algebra, World Scientific, page 124

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   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.

   2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114

   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;.

   2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1

2. field theoryalgebraA polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.

   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.

   2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69

   2 more examples

   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.

   2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145

   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.

   2008, Stephen D. Cohen, Mateja Preŝern, “The Hansen-Mullen Primitivity Conjecture: Completion of Proof”, in James McKee, Chris Smyth, editors, Number Theory and Polynomials, Cambridge University Press, page 89