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?



on the computation of decomposition groups
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.


