Given a field F and an extension field K⊇F, if α element K is an algebraic element over F, the minimal polynomial of α over F is the unique monic irreducible polynomial p(x) element F[x] such that p(α) = 0. It is the generator of the ideal {f(x) element F[x]|f(α) = 0} of F[x]. Any irreducible monic polynomial p(x) of F[x] has some root α in some extension field K, so that it is the minimal polynomial of α. This arises from the following construction. The quotient ring K = F[x]/〈p(x)〉 is a field, since 〈p(x)〉 is a maximal ideal, moreover K contains F. Then p(x) is the minimal polynomial of α = x^_, the residue class of x in K.

algebraic number minimal polynomial | conjugate elements | matrix minimal polynomial