From: Bill Dubuque <wgd@berne.ai.mit.edu>
Subject: Re: Abhyankhar's Lemma
Date: 24 Jan 1999 09:35:58 -0500
Newsgroups: sci.math

Roberto Maria Avanzi <mocenigo@werner.exp-math.uni-essen.de> wrote:
| 
| What is Abhynkhar's Lemma ?
| Where do I find it precisely ?
| Does anybody know the statement ?
| 
| I am asking a lot of persons about it, and all they tell me is
| "look in a basic book about Algebraic Number Theory".
| Fact is, I found it used in some papers of the 70s as well as
| in a book of Serre.  With absolutely NO references, no statement
| of the result.  I understand it deals with the behaviour of
| ramification in composita of fields, but I cannot infer the
| statement from the context...

ABHYANKAR'S LEMMA  Let  L = L_1 L_2  be the composite of two finite
algebraic extension fields  L_i  of  K, let  P  be a prime divisor
of  L, which is ramified in  L_i|K  of order  e_i (i=1,2); then, if  
e_2|e_1  and  P  is tame in  L_2|K, then  P  is unramified in  L|L_1.

(cf. MR 51#5553 [1]; see also [2]).

-Bill Dubuque

[1] Ishida, Makoto.  Class numbers of algebraic number fields of Eisenstein 
type. II.  J. Number Theory 6 (1974), 99-104. MR 51 #5553

[2] Gold, Robert; Madan, M. L.  Some applications of Abhyankar's lemma.
Math. Nachr. 82 (1978), 115--119. MR 58 #5601 (Reviewer: W.-D. Geyer)
==============================================================================

From: spamkill.lahtonen@utu.fi (Jyrki Lahtonen)
Subject: Re: Abhyankhar's Lemma
Date: 25 Jan 1999 08:09:44 GMT
Newsgroups: sci.math

In article <Pine.A32.3.95.990118120123.27556A-100000@werner.exp-math.uni-essen.de>, Roberto Maria Avanzi <mocenigo@werner.exp-math.uni-essen.de> says:
>
>What is Abhynkhar's Lemma ?
>Where do I find it precisely ?
>Does anybody know the statement ?
>
>I am asking a lot of persons about it, and all they tell me is
>"look in a basic book about Algebraic Number Theory".
>Fact is, I found it used in some papers of the 70s as well as
>in a book of Serre.  With absolutely NO references, no statement
>of the result.  I understand it deals with the behaviour of
>ramification in composita of fields, but I cannot infer the
>statement from the context...
>
>Thank you in advance
> Roberto
>
My only encounter with Abhyankar's Lemma is from a recent book
"Algebraic function fields and codes" by H.Stichtenoth (Springer,
Universitext series). There it's phrased as follows:

Let F'/F be a finite separable extension of function fields. Suppose
that F'=F_1 F_2 is a compositum of two intermediate fields F_1 and F_2.
Let P be a place of F and P' a place of F' lying above P. Set 
P_i= F_i \cap P' for i=1,2. Assume that at least one of the extensions
P_i|P is tame (i.e. the ramification index e(P_i|P) is relatively prime
to the characteristic of F). Then 

                e(P'|P)= lcm(e(P_1|P), e(P_2|P)).

It is relatively easy to find a counterexample in the case that both
extensions F_i/F are wildly ramified at P. I don't have one at hand,
but could easily look it up from my seminar notes, if you are interested.

I don't know, if there are variations of this theme in number fields
or elsewhere.

Jyrki Lahtonen, Ph.D.
Department of Mathematics,
University of Turku,
FIN-20014 Turku, Finland
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