Extension Field Minimal Polynomial

Given a field and an extension field , if is an algebraic element over , the minimal polynomial of over is the unique monic irreducible polynomial such that . It is the generator of the ideal

of .

Any irreducible monic polynomial of has some root in some extension field , so that it is the minimal polynomial of . This arises from the following construction. The quotient ring is a field, since is a maximal ideal, moreover contains . Then is the minimal polynomial of , the residue class of in .

, which is also the simple extension field obtained by adding to . Hence, in this case, and the extension field coincides with the extension ring.

In general, if is any other algebraic element of any extension field of with the same minimal polynomial , it remains true that , and this field is isomorphic to .