English Dictionary

primitive polynomial

noun

(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.

A primitive polynomial is a polynomial that cannot be factored into polynomials of lower degree over a given field.

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].

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.