Galois Extension: Proving L is Galois Over K

[math_grl]

This stuff is killing me...

Let K \leq M \leq L be fields such that L is galois over M and M is galois over K. We can extend \phi \in G(M/K) to an automorphism of L to show L is galois over K.

I need help filling in the details in why exactly L is galois over K.

 

[Tinyboss]

Something like this, I think: let [M:K]=a and [L:M]=b. Since M/K is Galois, there are a automorphisms of M that fix K. For the same reason, there are b automorphisms of L that fix M. So there are ad automorphisms of L that fix K. Since ad is also the degree of L/K, it's Galois.

If you're using Dummit and Foote, check out theorems 13.8 and 13.27.