primitive polynomials

Noun

primitive polynomial (plural primitive polynomials)

1. (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 #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;.1992, T. T. Moh, Algebra, World Scientific, page 124
2. (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.2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69