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?
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a. Let K be a field of characteristic p > 0 and let c E 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.
We have:
f(alpha+1) = (alpha+1)^p - (alpha+1) - c
= alpha^p + 1 - alpha - 1 - c
= alpha^p - alpha - c = 0
Whence alpha + 1 is a root of f whenever alpha is a root of f. Thus we see that if f has at least one root, then it has p roots, i.e. it splits over K, in which case Gal(K(alpha)/K) is trivial. On the other hand, if f is irreducible over K, then ...
Solution Summary
The expert solves two problems in Galois theory. Which subfield generates over Q is determined.