primitive polynomial
Language: English
Part of speech: noun
Forms: form: primitive polynomials
Senses: examples: text: 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;.
ref: 1992, T. T. Moh, Algebra, World Scientific, page 124:
text: 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.
ref: 2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114:
text: 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;.
ref: 2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1:
topics: algebra
mathematics
sciences
qualifier: ring theory
glosses: 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.
examples: text: 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.
ref: 2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69:
text: 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.
ref: 2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145:
text: 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.
ref: 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:
topics: algebra
mathematics
sciences
qualifier: field theory
glosses: A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
Hyponyms: sense: polynomial over an integral domain such that no noninvertible element divides all of its coefficients
word: monic polynomial
Related terms: english: of a polynomial
translation: of a polynomial
word: primitive part
english: of a polynomial
translation: of a polynomial
word: content
word: irreducible
word: primitivity