Suppose that is an irreducible polynomial over some field Let be a finite field extension. The following problem investigates the irreducibility of over

**Problem**. Suppose that is a finite field extension and Prove that if is irreducible over and then is irreducible over

**Solution**. We may assume, without loss of generality, that is monic. Let be a root of in some extension of and let be the minimal polynomial of over Clearly

We also have

It now follows from and that and so by

Let Then by and the fact that and are monic. We also have and thus by the minimality of Hence

Note that it is possible for to be irreducible over but For example consider and

Advertisements

very helpful result…