Galois structure in weakly ramified extensions of \(\mathbb Q\)
type: Item
instance of: scholarly article
title: Galois structure in weakly ramified extensions of \(\mathbb Q\)
zbMATH Open document ID: 0997.11098
author: Stéphane Vinatier
published in: Journal of Number Theory
publication date: 2002-08-25T00:00:00Z
review text: This paper provides a freeness result for a suitable Galois module attached to so-called weakly ramified Galois extensions \(N\) of the rationals. We recall here that for tamely ramified extensions \(N/K\) the obvious candidate is the Galois module \(O_N\), but \(O_N\) is not locally free over \({\mathbb Z}[G]\) if \(N/K\) is not tame. It turns out however that there is a nice substitute for \(O_N\) in case \(N/K\) is weakly ramified of odd degree, to wit: the square root \(A_{N/K}\) of the inverse different [\textit{B. Erez}, Math. Z. 208, 239-255 (1991; Zbl 0737.11032)]. We have to explain terminology now: As is well known, \(N/K\) is tamely ramified at a prime \(\mathbf p\) iff the ramification exponent \(e_{\mathbf p}\) is prime to \(\mathbf p\), and \(N/K\) is wildly ramified if and only if it is not tame. Moreover, \(N/K\) is declared to be weakly ramified at \(\mathbf p\) if the second ramification group \(G_2({\mathbf p})\) is trivial. Theorem 1.1 gives a neat classification of abelian wildly and weakly ramified extensions of \({\mathbb Q}_p\): they are all gotten as subfields of the compositum of \(L\) and an unramified extension of \({\mathbb Q}_p\), where \(L\) is the degree \(p\) subfield of \({\mathbb Q}_p(\zeta_{p^2})\). At first glance this might be surprising because this forces the ramification exponent to be \(p\) (totally wild ramification); it cannot be \(p(p-1)\) for example, and indeed as the author points out, \({\mathbb Q}_p(\zeta_{p^2})\) is not weakly ramified over \({\mathbb Q}_p\). This seems to indicate that the class of weakly ramified extensions is in some way not very large. The main result of the paper (Theorem 1.2) states that if \(N/\mathbb Q\) is any weakly ramified Galois of odd degree and if all decomposition groups at wild places are abelian, then the locally free \({\mathbb Z}[G]\)-module \(A_{N}\) is free. The proof uses the Hom description of the class group, Gauss sum and resolvents, plus a fair amount of explicit calculati…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 1787790
zbMATH Keywords: inverse different
zbMATH Keywords: Galois modules
zbMATH Keywords: wild ramification
describes a project that uses: Magma
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: jnth.2001.2674
OpenAlex ID: W4301223327
cites work: An efficient algorithm for the computation of Galois automorphisms
cites work: Integral Galois module structure of some Lubin-Tate extensions
cites work: Opérations d'Adams et groupe des classes d'algèbre de groupe. (Adams operations and class groups of group algebras)
cites work: Q4510764
cites work: Q4010507
cites work: The Galois structure of the square root of the inverse different
cites work: Q3808199
cites work: Q3966247
cites work: Q4132638
cites work: Q5094820
cites work: Q3859900
cites work: On Fröhlich's conjecture for rings of integers of tame extensions
cites work: On the Galois Module Structure of Rings of Integers of Wild Abelian Extensions
cites work: Une Famille Infinie d'Extensions Faiblement Ramifiées
DOI: 10.1006/JNTH.2001.2674
description: scientific article; zbMATH DE number 1787790