Question:
Problems in Galois Theory
a. Let K be a field of characteristic p > 0, and let c in K. Show that if alpha is a root of f (x) = x^p - x - c, so is alpha + 1. Prove that K(alpha) is Galois over K with group either trivial or cyclic of order p.
b. Find all subfields of Q ( sqrt2, sqrt 3) with proof that you have them all. What is the minimal polynomial of sqrt2+ sqrt3? Which subfields does it generate over Q?