primitive polynomial

英語（米国） (en-US) 英語（イギリス） (en-GB)

名詞

(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.

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Gauss's lemma implies that a primitive polynomial in R[x] that factors over the field of fractions must factor as a product of primitive polynomials in R[x].

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意味（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.

We claim that every primitive polynomial can be written as a product of irreducible elements in mathbf D[x]. […] 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 mathbf D[x].

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意味（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.

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.

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primitive polynomials （ plural ）