Kummer's criterion over global function fields
type: Item
instance of: scholarly article
title: Kummer's criterion over global function fields
zbMATH Open document ID: 0816.11036
author: Linghsueh Shu
published in: Journal of Number Theory
publication date: 1995-02-16T00:00:00Z
review text: Let \(k\) be a global function field (i.e., a finite extension of a rational function field \(\mathbb{F}_ p (T)\) over the finite field with \(p\) elements) and \(\infty\) a fixed place of degree one over the field \(\mathbb{F}_ q\) of constants. Let further \(A\) be the Dedekind ring of elements of \(k\) regular off \(\infty\), and \({\mathfrak p}\) a (non-zero) prime ideal. With \({\mathfrak p}\), one can associate an abelian extension \(k({\mathfrak p})\) of \(k\), which is similar to the field of \(p\)-th roots of unity \(k(p):= \mathbb{Q}(\zeta_ p)\) over \(k:= \mathbb{Q}\). We have \(k\subset H_ \infty \subset k^ + ({\mathfrak p}) \subset k({\mathfrak p})\), where \(H_ \infty\) is the ring class field of \(A\) and \(k^ + ({\mathfrak p})\) the ``maximal real subfield of \(k({\mathfrak p})\)'', i.e., the maximal abelian extension of \(k\) unramified off \({\mathfrak p}\) and completely split at \(\infty\). The Galois groups are as follows (where \(C\ell\) means ``class group''): \[ C\ell(A) {\overset \cong \longrightarrow} \text{Gal} (H_ \infty/k), \qquad (A/{\mathfrak p})^* {\overset \cong \longrightarrow} \text{Gal} (k({\mathfrak p})/k), \qquad \mathbb{F}_ q^* {\overset \cong \longrightarrow} \text{Gal} (k({\mathfrak p}) /k^ +({\mathfrak p})). \] In this situation, there are several natural Galois modules: divisor class groups of \(k({\mathfrak p})\), \(k^ +({\mathfrak p})\), \(H_ \infty\), of their respective rings of \(A\)-integers, groups of units, of cyclotomic units etc. The connection between these objects has been studied by Galovich-Rosen, Goss, Hayes, Okada, Oukhaba, and others. In particular, in the case where \(k= \mathbb{F}_ q(T)\) and \(\infty\) is the usual infinite place, D. Goss proved a useful criterion for \(C\ell(O^ +({\mathfrak p}))\) to have a non-trivial \(p\)-part [cf. \textit{D. Goss}, The arithmetic of function fields; de Gruyter, 313-402 (1992; Zbl 0806.11028); \(O^ + ({\mathfrak p})= A\)-integers in \(k^ +({\mathfrak p})\)]. It is an…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 710375
zbMATH Keywords: global function field
zbMATH Keywords: class groups
zbMATH Keywords: unit indices
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
full work available at URL: jnth.1994.1098
OpenAlex ID: W2061672032
DOI: 10.1006/JNTH.1994.1098
description: scientific article; zbMATH DE number 710375