You're missing some information on . For example if K is the splitting field for f then as it is a field extension of . So I think the theorem will hold if you maintain that .
Consider the two finite fields and and let n be a prime number, q be a generator of .
Now show that the polynomial is irreducible over -->
Well if I define then I think I'll have to assume that f(X)=h(X)*g(X) over with degree(h)>0 and degree(g)>0 ? But how does one do that?
q has order n-1. What does that mean? Doesn't that mean that f(q) has order n-1 as well? How can I prove this?