I'm trying to make sense of all these theorems related to ramification. I was hoping someone would summarize these results. Assume we have:
-
An extension $L/K$ and a some subextensions $E_1,\ldots,E_k$ sitting in between (without necessarily having $E_i\leq E_j$ or vice versa for pairs).
-
What about if some of these extensions are Galois?
I was wondering what ramification of a prime in $L/K$ tells us about ramification for the $E_i/K$ or $L/E_i$ and vice versa? How much can we pin down where the ramification has to be in these subextensions or do we always have to just compute discriminants?
I'm also trying to figure out how to compute Hilbert class fields. This is why I'm interested in how given some field $K/\mathbb{Q}$ I should go about trying to find a field $L/K$ that kills all ramification? I should be able to somehow figure out what's allowed to ramify in $L/\mathbb{Q}$?
EDIT: This is one argument that I just don't get. The Hilbert Class Field of $\mathbb{Q}(\sqrt{-5})$ is $\mathbb{Q}(\sqrt{-5},\sqrt{-1})$. Here's the standard argument:
$\mathbb{Q}(\sqrt{5})/\mathbb{Q}$ is unramified outside of 5 and $\mathbb{Q}(\sqrt{-1})/\mathbb{Q}$ is unramified outside 2. Hence, $\mathbb{Q}(\sqrt{-5},\sqrt{-1})/\mathbb{Q}(\sqrt{-5})$ is unramified everywhere.
Why can't there be some prime of $\mathbb{Q}(\sqrt{-5})$ that ramifies and ramifies above the other two extensions?
Best Answer
$ \newcommand{\fp}{{\mathfrak p}} \newcommand{\fP}{{\mathfrak P}} $ The ramification in intermediate fields is a slightly delicate (purely group theoretic) issue. First, assume without loss of generality that $L/K$ is Galois with Galois group $G$, otherwise pass to the Galois closure and repeat the following argument for $L$ being the intermediate field.
Suppose that you have a prime $\fp$ of $K$ with prime $\fP$ of $L$ lying above $\fp$. Let $D=D_{\fP/\fp}$ be the decomposition group, i.e. the subgroup of $G$ consisting of elements that fix the place $\fP$. Inside that, you have the inertia subgroup $I=I_{\fP/\fp}$. Recall that the index of $D$ in $G$ is the number of primes in $L$ lying above $\fp$, and $|I|$ is the ramification index of $\fP/\fp$. The residue field degree is the index of $I$ in $D$. Now, let $E$ be an intermediate field, corresponding to a subgroup $H$ of $G$, so that $H=\text{Gal}(L/E)$. Then
Exercise 1: There is a natural bijection between double cosets $H\backslash G/D$ and the primes of $E$ lying above $\fp$. (You might want to begin by checking that this is independent of the choice of $\fP$. Another prime above $\fp$ would have given a conjugate decomposition group.)
Exercise 2: If, under the above correspondence, the double coset $HgD$ corresponds to the prime $\fp'$ of $L$, then $$ D_{\fP/\fp'} = H\cap D^g\leq H\text{ and } I_{\fP/\fp'} = H\cap I^g\leq H. $$ This allows you to compute the ramification index and the residue field degree of $\fP/\fp'$ as above.
I hope that that answers your questions. As for explicit computation of Hilbert class fields, you might want to consult Sections 3 and 4 of Cohen's Advanced Topics in Computational Number Theory - a wonderful book that treats your question in great detail.