primitive polynomialとは 意味・読み方・使い方

名詞

primitive polynomial (複数形 primitive polynomials)

1. (algebra, ring theory) 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.
   • 1992, T. T. Moh, Algebra, World Scientific, page 124,

     We claim that every primitive polynomial can be written as a product of irreducible elements in ${\displaystyle \mathbf {D} [x]}$. […] By induction on the degree of the primitive polynomials, we conclude that both ${\displaystyle g(x),h(x)}$ can be written as product of irreducible elements in ${\displaystyle \mathbf {D} [x]}$.

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

     If ${\displaystyle f(x)\in \mathbb {Z} [x]}$, the ring of polynomials with coefficients in ${\displaystyle \mathbb {Z} }$, then the content of ${\displaystyle f}$, denoted by ${\displaystyle c(f)}$, is the greatest common divisor of the coefficients of ${\displaystyle f}$. The polynomial ${\displaystyle f(x)}$ is called a primitive polynomial if ${\displaystyle c(f)=1}$. Since ${\displaystyle c(fg)=c(f)c(g)}$, by Gauss's lemma [Hungerford, 74], the set ${\displaystyle S}$ of primitive polynomials in ${\displaystyle \mathbb {Z} [x]}$ is a multiplicatively closed set. Define ${\displaystyle \Lambda =\mathbb {Z} [x]_{S}}$, the localization of ${\displaystyle \mathbb {Z} [x]}$ at ${\displaystyle S}$, a subring of the field of quotients ${\displaystyle \mathbb {Q} (x)}$ of ${\displaystyle \mathbb {Z} [x]}$. Elements of ${\displaystyle \Lambda }$ are of the form ${\displaystyle f(x)/g(x)}$ with ${\displaystyle f(x),g(x)\in \mathbb {Z} [x]}$ and ${\displaystyle g(x)}$ a primitive polynomial.

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

     According to the Gauss lemma, the product of primitive polynomials is primitive. Therefore if ${\displaystyle f(x),g(x)}$ are primitive and ${\displaystyle f(x)=g(x)h(x)}$ with ${\displaystyle h(x)}$ in ${\displaystyle F[x]}$, then necessarily ${\displaystyle h(x)}$ is in ${\displaystyle A[x]}$ and primitive. […] The irreducible elements of ${\displaystyle A[x]}$ are irreducible elements of ${\displaystyle A}$ and primitive polynomials which are irreducible in ${\displaystyle F[x]}$.

2. (algebra, field theory) A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
   • 2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69,

     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.

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

     Definition 7.2 Let ${\displaystyle f(x)}$ be a monic polynomial of degree ${\displaystyle n}$ over ${\displaystyle \mathbb {F} _{q}}$. If ${\displaystyle f(x)}$ has a primitive element of ${\displaystyle \mathbb {F} _{q^{n}}}$ as one of its roots, ${\displaystyle f(x)}$ is called a primitive polynomial of degree ${\displaystyle n}$ over ${\displaystyle \mathbb {F} _{q}}$.

     Theorem 7.7 For any positive integer ${\displaystyle n}$ there always exist primitive polynomials of degree ${\displaystyle n}$ over ${\displaystyle \mathbb {F} _{q}}$. All the ${\displaystyle n}$ roots of a primitive polynomial of degree ${\displaystyle n}$ over ${\displaystyle \mathbb {F} _{q}}$ are primitive elements of ${\displaystyle \mathbb {F} _{q^{n}}}$. All primitive polynomials of degree ${\displaystyle n}$ over ${\displaystyle \mathbb {F} _{q}}$ are irreducible over ${\displaystyle \mathbb {F} _{q}}$. The number of primitive polynomials of degree ${\displaystyle n}$ over ${\displaystyle \mathbb {F} _{q}}$ is equal to ${\displaystyle \phi (q^{n}-1)/n}$.

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

     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.