Recognizing simple subextensions of purely transcendental field extensions
type: Item
instance of: scholarly article
title: Recognizing simple subextensions of purely transcendental field extensions
zbMATH Open document ID: 0997.12006
author: Joern Mueller-Quade
author: Rainer Steinwandt
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2002-06-18T00:00:00Z
review text: Let \(k(X)/k\) be a finitely generating purely transcendental field extension, where \(X\) is algebraically independent over \(k\). It is known that all intermediate fields \(K\) of \(k(X)/k\) with \(\text{transdeg} (K/k)=1\) are simple extensions of \(k\). In this paper, the authors give a criterion that allows one to decide effectively whether the extension \(K/k\) is simple if a finite generating set of \(K\) over \(k\) is known and computations in \(K\) are effective. In the affirmative case, a primitive element of \(K\) over \(k\) is computed. Let \(Z= \{Z_x\mid x\in X\}\), where \(Z_x\) is transcendental over \(K\). Let \(\phi\) be the homomorphism of \(K[Z]\) into \(k(X)\) such that \(\phi(Z_x)= x\) for all \(x\in X\) and \(\phi\) is the identity on \(K\). Let \(B\) denote the kernel of \(\phi\). Let \(G\) be a reduced Gröbner basis of \(B\). Then the authors show that the following statements are equivalent: (i) \(K/k\) is simple. (ii) \(B\) is principal. (iii) \(G= \{n(Z)- fd(Z)\}\) for some \(n(Z),d(Z)\in k[Z]\), \(f\in K\) or \(G= \emptyset\). If (iii) holds and \(G\neq \emptyset\), then \(f\) is a primitive element of \(K/k\). If \(G= \emptyset\), then \(K=k\). The authors show that (i) and (ii) are equivalent if the assumption that \(X\) is finite is dropped.
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1523456
zbMATH Keywords: rational function fields
zbMATH Keywords: primitive element
zbMATH Keywords: Gröbner bases
zbMATH Keywords: purely transcendental field extension
zbMATH Keywords: intermediate fields
zbMATH Keywords: simple extensions
reviewed by: John N. Mordeson
MaRDI profile type: Publication
full work available at URL: s002000000024
OpenAlex ID: W2028786182
DOI: 10.1007/S002000000024
description: scientific article; zbMATH DE number 1523456