On a theorem of M. Deuring and I. R. Šafarevič
type: Item
instance of: scholarly article
title: On a theorem of M. Deuring and I. R. Šafarevič
zbMATH Open document ID: 0369.12011
DOI: 10.1007/BF01168587
author: Manohar L. Madan
published in: Manuscripta Mathematica
publication date: 1977-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/154509
review text: Let \(F/K\) be a field of algebraic functions of one variable, \(K\) algebraically closed of characteristic \(p\neq 0\). Let \(E/K\) be a cyclic extension of \(F/K\) of degree \(\ell\), a prime not necessarily different from \(p\). Let \(\xi\), \(\rho\) denote the \(\ell\)-ranks of the null class groups of \(E\) and \(F\) respectively. If \(E/F\) is ramified, \textit{M. Deuring} [Math. Ann. 113, 208--215 (1936; Zbl 0014.29301)] proved that \(\xi=\ell\rho+(t-\delta)(\ell-1)\) where \(t\) is the number of ramified primes and \(\delta\) is 1 or 2 according as \(\ell\) equals \(p\) or not. \textit{I. R. Shafarevich} [Am. Math. Soc., Transl., II. Ser. 4, 59--72 (1956); translation from Mat. Sb., N. Ser. 20(62), 351--363 (1947; Zbl 0041.17101)] proved the same relation in the unramified case for \(\ell=p\) where \(\xi\), \(\rho\) denote the ranks of the Hasse-Witt matrices. \textit{D. Subrao} [Manuscr. Math. 16, 169--193 (1975; Zbl 0321.14017)] gave a unified proof for \(\ell=p\). \textit{M. Rosen} [J. Number Theory 9, 160--174 (1977; Zbl 0371.12013)] and \textit{F. J. Sullivan} [Arch. Math. 26, 253--261 (1975; Zbl 0331.14016)] have proved the theorem in special cases. In this paper, using Galois cohomology, Deuring's method of proof is modified so as to give a proof of the theorem in all cases.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 3575702
zbMATH Keywords: field of algebraic functions of one variable
zbMATH Keywords: \(\ell\)-ranks of the null class groups
zbMATH Keywords: Galois cohomology
zbMATH Keywords: modified Deuring's method of proof
MaRDI profile type: Publication
cites work: Class number and ramification in fields of algebraic functions
cites work: Ambiguous divisor classes in function fields
cites work: The p-rank of Artin-Schreier curves
cites work: \(p\)-torsion in the class group of curves with too many automorphisms
cites work: Q4754855
OpenAlex ID: W2001476362
description: scientific article; zbMATH DE number 3575702