A primitive polynomial is a polynomial that generates all elements of an extension field from a base field. Primitive polynomials are also irreducible polynomials. For any prime or prime power q and any positive integer n, there exists a primitive polynomial of degree n over GF(q). There are a_q(n) = (ϕ(q^n - 1))/n primitive polynomials over GF(q), where ϕ(n) is the totient function. A polynomial of degree n over the finite field GF(2) (i.e., with coefficients either 0 or 1) is primitive if it has polynomial order 2^n - 1.

finite field | irreducible polynomial | polynomial | polynomial order | primitive element | primitive root