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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?

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2 Answers

up vote 2 down vote accepted

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.

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I checked that, but it seems that Henri Cohen assumed in his book that all those were built on number fields. What I really want to do is to compute all these in function fields, and I don't think the algorithms in number fields can be just used on function fields :( –  Yujia Qiu Jun 11 '10 at 9:07
    
Dedekind's theorem, which is the basis for most of Cohen's algorithms in these chapters, is still valid. The same is true for the resolvents in 6.3. For example, computing the maximal order becomes much easier since extracting the square free part of the discriminant is polynomial (gcd with derivative). In fact, most of the algorithms, after a suitable change to function fields, usually become easier. On a bigger scale than the original question, the ray class groups also become easier to calculate (Kedlaya/Satoh's algorithms), so you can actually compute using Artin symbols. –  Dror Speiser Jun 11 '10 at 10:38
    
Thanks a lot, Dror –  Yujia Qiu Jun 11 '10 at 13:40
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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.

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Thanks Hagen. The extension you mentioned is basically Kummer extension, right? I found that the article is in German, is there any English translation of this? I am afraid that my limited German is not enough to read an article :P –  Yujia Qiu Jun 8 '10 at 15:20
    
Yes the starting point is Kummer theory, but then one has to discuss specific transformations of generators of $L/k(x)$ to bring the polynomial $f(x)$ into a certain form. As for an english translation of Hasse's article: I do not know of any such translation - sorry. Hagen –  Hagen Jun 9 '10 at 8:15
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