Extension Field Minimal Polynomial

Given a field F and an extension field K superset= F, if alpha in K is an algebraic element over F, the minimal polynomial of alpha over F is the unique monic irreducible polynomial p(x) in F[x] such that p(alpha)=0. It is the generator of the ideal

of F[x].

Any irreducible monic polynomial p(x) of F[x] has some root alpha in some extension field K, so that it is the minimal polynomial of alpha. 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 alpha=x^_, the residue class of x in K.

K=F[x^_]=F[alpha], which is also the simple extension field obtained by adding alpha to F. Hence, in this case, F[alpha]=F(alpha) and the extension field coincides with the extension ring.

In general, if beta is any other algebraic element of any extension field of F with the same minimal polynomial p(x), it remains true that F[beta]=F(beta), and this field is isomorphic to F[x]/<p(x)>.

See also

This entry contributed by Margherita Barile

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Barile, Margherita. "Extension Field Minimal Polynomial." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExtensionFieldMinimalPolynomial.html

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