
Galois Theory
Okay i have these two problems that can't solve so I decided to share:
1.Let f(x) be an irreducible polynomial over F of degree n, and let K be a field extension of F with [K:F]=m. If gcd(n,m)=1 then f is irreducible over K
2.Let K be a normal extension of F, and let f(x) be an irreducible polynomial over F (in F[x]). Let p, q be two monic irreducible factors of f in K. Prove that there is an h in Gal(K/F) such that h(p)=q. (h denotes both the Fautomorphism of K and the natural induced homomorphism in K[x])
In the first one, I get that K has no root of K, and so f has no linear factors over K; but I can't see the general case.
In the second I tried building the splitting field of f over K, then since the elements of the Galois group permute the roots of f I try to translate that to K, but I get lost on how to proceed.
Thanks in advance anway.
Edit: Okay, I think I got the first one. But for the second one I'm stuck, I'm I have to use the isomorphism extension theorem: since p, q are irreducible and monic they are the minimal polynomial of some element (actually all, since we define the extension as the splitting field of p and q) and we can choose the isomorphism extension to send that root to any root of the image of the polynomial p or q (depending on which you chose the original root from) under the base isomorphism. This is where I am, I need (or so I think) to send the roots of p to those of q and also the other way around (Notice that the existence of a function h that satisfies the conditions implies deg(p)=deg(q)), but I don't really have any idea.
Thanks again.