Finite radical field extensions and crossed homomorphisms
type: Item
instance of: scholarly article
title: Finite radical field extensions and crossed homomorphisms
zbMATH Open document ID: 0864.12002
author: Florin Nicolae
author: Toma Albu
published in: Journal of Number Theory
publication date: 1997-06-22T00:00:00Z
review text: Let \(K^*\) denote the multiplicative group of nonzero elements of a field \(K\), and call \(K\) a radical extension of a subfield \(F\) if \(K=F(\Delta)\) for a subgroup \(\Delta\) of \(K^*\) such that \(\Delta\) contains \(F^*\) and \(\Delta/F^*\) is a torsion group. Now let \(E\) be a finite Galois extension of \(F\) with Galois group \(G\), and let \(T(E/F)\), respectively \(\mu (E)\), be the subgroup of all elements \(x\) of \(E^*\) such that \(x^n\in F^*\), respectively \(x^n=1\), for some positive integer \(n\). By expanding concepts from Kummer theory, the authors establish an isomorphism between \(T(E/F)/F^*\) and the group of crossed homomorphisms, i.e. one cocycles, of \(G\) into \(\mu(E)\). This isomorphism is used to characterize the subfields of \(E\) which are radical extensions of \(F\). If \(L\) is a finite field extension of \(F\) such that \(L\cup E=F\) and \(\mu(LE) =\mu(E)\), this characterization is employed to obtain a one-to-one correspondence between the subfields of \(E\) which are radical extensions of \(F\) and the subfields of \(LE\) which are radical extensions of \(L\). The authors obtain several more technical applications of this characterization.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 955260
zbMATH Keywords: Galois extension
zbMATH Keywords: Kummer theory
zbMATH Keywords: group of crossed homomorphisms
zbMATH Keywords: one cocycles
zbMATH Keywords: radical extensions
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
full work available at URL: jnth.1996.0124
OpenAlex ID: W2028947384
DOI: 10.1006/JNTH.1996.0124
description: scientific article; zbMATH DE number 955260