Describing Galois groups of some local fields

Question

We can describe the Galois group of some global fields explicitly, for example, we can describe the Galois group of splitting field of $x^n-a$ over the rationals explicitly, especially the cyclotomic fields.

Suppose that $K/F$ is an extension of number fields, and $\mathfrak{P}|\mathfrak{p}$ are primes in these extensions. I know that decomposition groups of a prime $\mathfrak{p}$, are isomorphic to $Gal(K_{\mathfrak{P}}/F_\mathfrak{p})$.

Also, I know that for any positive integer $t$, there is a unique unramified extension of degree $t$ over $F_\mathfrak{p}$, which can be constructed by adjoining the $(q^t-1)^{th}$ roots of unity, and its Galois group is cyclic.

Even in this case I can not describe the $Gal(F_\mathfrak{p}(\zeta_{q^t-1})/F_\mathfrak{p})$ explicitly, I don't even know it well enough. Could you give some insights on these Galois groups?

Also I have no idea about other similar extensions, something like $Gal(F_\mathfrak{p}(\sqrt[n]a)/F_\mathfrak{p})$.

Answer 1

I will address the unramified extensions and leave it to someone else to discuss some tricks on calculating Galois groups explicitly. But briefly, the Galois theory here is much easier than in the number field case since there is only one prime to deal with and in every extension only two things happen: The residue field grows and/or the valuation becomes finer.

The structure of unramified extensions of $\mathbb Q_p$ perfectly mirrors the structure of field extensions of $\mathbb F_p$. Some details below:

Let $p$ be a prime number and $q = p^n$. As you probably know, the units of the field $\mathbb F_{q^t}$ form a cyclic group and therefore are exactly the $q^t-1$th roots of unity. Moreover, the Galois group Gal$( \mathbb F_{q^t}/\mathbb F_q)$ is cyclic and is generated by the $q$-Frobenius, i.e. the map that takes $x \to x^q$.

Now there is a construction called the ring of Witt vectors that takes a finite field $\mathbb F_q$ (more generally a perfect field of characteristic $p>0$) and produces a complete local ring of characteristic $0$ called $W(\mathbb F_q)$ which actually turns out to be a very familiar ring to us! For every positive integer $n$, there is a unique unramified extension of $\mathbb Q_p$ (call it $K_n$) and it's valuation ring is exactly $W(\mathbb F_q)$ and there is a canonical isomorphism $$Gal\left( K_n/\mathbb{Q_{p}} \right) \simeq Gal \left(\mathbb F_q/\mathbb F_p\right)$$

So what's the point? The Galois theory of unramified extensions of local fields is as easy and explicit as the Galois theory of finite fields.

I would start playing around with $\mathbb Q_p$ and adjoining roots of small degree irreducible polynomials and understand what's going on in each case with the help of Hensel's lemma.