From: Bill Dubuque
Subject: Re: Abhyankhar's Lemma
Date: 24 Jan 1999 09:35:58 0500
Newsgroups: sci.math
Roberto Maria Avanzi 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_iK of order e_i (i=1,2); then, if
e_2e_1 and P is tame in L_2K, then P is unramified in LL_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), 99104. MR 51 #5553
[2] Gold, Robert; Madan, M. L. Some applications of Abhyankar's lemma.
Math. Nachr. 82 (1978), 115119. MR 58 #5601 (Reviewer: W.D. Geyer)
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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 , Roberto Maria Avanzi 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_iP is tame (i.e. the ramification index e(P_iP) is relatively prime
to the characteristic of F). Then
e(P'P)= lcm(e(P_1P), e(P_2P)).
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,
FIN20014 Turku, Finland
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