primitive polynomial

(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.

Japanese Meaning

原始多項式：整域 R 上の多項式のうち、全ての係数に共通して非可逆元が割り切られることがなく、すなわち係数の最大公約数が1である多項式。 / より一般的には、GCD整域 R 内で、各係数に共通する非単元（非単位元）によって全体の割り切りを受けない多項式を指す。

What is this buttons?

Google Translation / DeepL Translation

ガウスの補題は、R[x] の原始多項式が分数体上で因数分解されるならば、R[x] においても同じ性質を持つ原始多項式の積として因数分解されることを示す。

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Sense（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].

Google Translation / DeepL Translation

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Sense（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.

Google Translation / DeepL Translation

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primitive polynomials

（ plural ）

Details

primitive polynomial

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