Galois groups of \(p\)-extensions with two ramification points
type: Item
instance of: scholarly article
title: Galois groups of \(p\)-extensions with two ramification points
zbMATH Open document ID: 0985.11057
DOI: 10.1007/BF02465125
author: H. Markšaitis
published in: Lithuanian Mathematical Journal
publication date: 2001-05-13T00:00:00Z
review text: Let \(p\) be an odd prime and \(q\) a prime with \(q\equiv 1\pmod {p^{r+1}}\), \(q\not\equiv 1\pmod{p^{r+2}}\), \(r>0\). The author studies the Galois group \(G\) of the maximal \(p\)-extension \(K\) of \(\mathbb{Q}\) unramified outside \(p\) and \(q\). He shows that \(G\) modulo fourth commutators is generated by two elements \(\sigma,\tau\) which are connected by one generating relation \(\sigma^{p^{r+1}} \equiv[\tau^{p^r} [\tau,\sigma]^a, \sigma]\). He gives an implicit determination of \(a\) by means of a unique subfield of \(K\). Furthermore, he determines \(a\) explicitly in the cases \(p=3\), \(q\in\{19,37\}\).
reviewed by: Helmut Koch
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 1551510
zbMATH Keywords: maximal \(p\)-extension
zbMATH Keywords: Galois group
MaRDI profile type: Publication
cites work: Q5730964
cites work: On Fields of Class Two
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cites work: A NOTE ON THE CLASS FIELD TOWER
cites work: On non‐ramified extensions with prescribed Galois group
cites work: l-Erweiterungen mit vorgegebenen Verzweigungsstellen.
cites work: Q5824718
cites work: Extensions with given points of ramification
full work available at URL: https://doi.org/10.1007/bf02465125
OpenAlex ID: W2022647311
description: scientific article; zbMATH DE number 1551510