Because if is not irreducible then is not necessarily a field!

And so it cannot be an extension field.

Here is one example. Say is primitive, then it means for some . If you want to compute all you need to do is compute the irreducible polynomial for over and define the automorphism where is a root of this polynomial in . Therefore, an easy way to see that is Galois is if and only if contains all the zeros of the minimal polynomial of with not repeated roots.2) Why primitive elements/polynomials are important?

Let be an algebraic extension.3) What does it means by conjugates. Here is a statement saying that

neither /alpha nor its conjugates, /alpha^j are elements to which /beta is

mapped.

Let . We say that are conjugates iff they have the same minimal polynomials over .