High Quality Content by WIKIPEDIA articles! In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF(pm). In other words, a polynomial F(X) with coefficients in GF(p) = Z/pZ is a primitive polynomial if it has a root in GF(pm) such that {0,1, alpha, alpha^2, alpha^3,dots,alpha^{p^{m}-2}} is the entire field GF(pm), and moreover, F(X) is the smallest degree polynomial having as root. In ring theory, the term primitive polynomial is used for a different purpose, to mean a polynomial over a unique factorization domain (such as the integers) whose greatest common divisor of its coefficients is a unit. This article will not be concerned with the ring theory usage. See Gauss's lemma. Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible.

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