Intersections of two cofinite subfields
type: Item
instance of: scholarly article
title: Intersections of two cofinite subfields
zbMATH Open document ID: 1054.12002
author: William C. Waterhouse
published in: Archiv der Mathematik
publication date: 2004-10-05T00:00:00Z
review text: Let \(A\) and \(B\) be subfields of a field \(F\). Let \(P\) denote the prime subfield of \(F\). Then it is known that it is possible for \(F\) to be infinite dimensional over \(A\cap B\) even if \(F/A\) and \(F/B\) are finite-dimensional. In this paper, the author shows that if \(F\) is finitely generated over \(P\), purely inseparable over \(A\) or \(B\), and finite-dimensional over both \(A\) and \(B\), then \(F\) is finite dimensional over \(A\cap B\). The author shows that \(F\) may be infinite dimensional over \(A\cap B\) if the assumption that \(F\) is finitely generated over \(P\) is removed. In fact, the author shows that if \(P\) is of characteristic \(p\) and \(F\) is generated over \(P\) by indeterminates \(x_1,x_2,\dots, y_1,y_2,\dots\), then there are subfields \(A\), \(B\), \(C\) with \(F\) Galois of degree 2 over \(A\) and purely inseparable of degree \(p\) over \(B\) and over \(C\), where \(F\) has infinite dimension over \(A\cap B\) and \(B\cap C\).
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 2106403
zbMATH Keywords: intersection of cofinite subfields
zbMATH Keywords: dimension
zbMATH Keywords: purely inseparable finitely generated field extension
zbMATH Keywords: automorphism
zbMATH Keywords: fixed field
reviewed by: John N. Mordeson
MaRDI profile type: Publication
full work available at URL: s00013-003-4872-0
OpenAlex ID: W1966688128
DOI: 10.1007/S00013-003-4872-0
description: scientific article; zbMATH DE number 2106403