a) $\mathbb{F}_q \subset \mathbb{F}_{q^n}$ where $q$ is a primary number
b) $\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}, \sqrt{3})$
c) $\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}, 5^{1/5})$
I know a field extension is Galois iff |Gal(E/F)| = [E:F], but is that the only way to tell? I can eaisily compute [E:F], but I feel like finding all the automorphisms is a bit annoying. I know for b it is Galois because be have a theorem about extensions of $\mathbb{Q}$ by square roots of distinct primes, but I don't see any other theorems that would apply for a or c.
I added a full attempt in the comments