on the computation of decomposition groups Let $L/K$ be a finite Galois extension of function fields, with Galois groups $G$. I want to look at the ramification of primes in the extension, i.e. to get $e_p$ and $f_p$ for a prime $p$ in the base field $K$ (since the extension is Galois, the ramification index and inertia degree are independent of the choice of the prime lying above $p$). From Serre's 'Local Fields', it is clear that if we fix a prime $q$ in $L$ which lies over $p$, then we can look at the decomposition group associated to $q$, say $G_q$, and its inertia group, say $(G_q)_0$ (please forgive me for the notation :P), and an immediate result is that $e_p = \left|{(G_q)_0} \right|$ and $f_p = \left| {G_q/(G_q)_0} \right|$.
And here is my problem. Is there any nice way to compute the decomposition groups, inertia groups or just the cardinalities? If not, can we do something in some special cases? For example, when $G$ is cyclic?
 A: Henri Cohen's A Course in Computational Algebraic Number Theory contains quite a bit of information. Chapters 4.8, 6.2 and 6.3 combined result in algorithms that compute decomposition groups. Note that if you want to relate different primes you will have to first compute the galois group (6.3) and fix a presentation.
A: A particularly well-studied case is that of cyclic extensions $L/K$, where $K=k(x)$ is a rational function field, the degree $n:=[L:K]$ is not divisible by the characteristic of $k$ and $k$ contains the $n$-th roots of unity. In this case there exists a kind of normal form $L=K(y)$ for generating $L$: $y^n=f(x)$, where the prime factorization of $f\in k[x]$ satisfies some requirements. One can then compute the ramification indices and inertia degrees using only the multiplicities and degrees of the prime factors of $f$. You can find the result in an article by Helmut Hasse: "Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper.", Journal für die reine und angewandte Mathematik 172 (1935).
Similar things can be done for Artin-Schreier-extensions of a rational function field.
Hagen.
