I found that I've never seriously studied about Galois theory of local fields, so I'm trying to find a good reference with some of (good) exercises. Here's a motivation:
When I studied abstract algebra, I did some computations about Galois groups of certain polynomials (over $\mathbb{Q}$, obviously), for degree 3 and (some of) degree 4 and 5 polynomials. I remember that Dummit & Foote's algebra is a very good references with good exercies. In case of local fields, I know that the Galois groups are solvable since we have a filtration of ramification groups. However, I don't have any experience of computing Galois groups of a given polynomial (over $\mathbb{Q}_p$, for example). For degree $\leq 3$, I think there shouldn't be a big difference. For example, in case of degree 3, Galois group of a splitting field of a given cubic polynomial is completely determined by squareness of a discriminant, and for local field we can use Hansel's lemma to determine whether a given discriminant is square or not (for $\mathbb{Q}$, it is much easier, if one have a sufficiently good computational power when discriminant is huge). The biggest difference should occur in degree 5, since $A_{5}$ is the smallest non-solvable group and it can't be a Galois group over $\mathbb{Q}_{p}$ or other local fields. A Galois group of a splitting field of $f(x) = x^{5}+20x + 16$ over $\mathbb{Q}$ is $A_{5}$, and basically I want to know what is a Galois group of it over $\mathbb{Q}_{p}$ (for all $p$). My guess is that computing Galois group would be much easier over local fields even for high degree polynomials, in contrast to over $\mathbb{Q}$.