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On a purely inseparable extension of a normal extension of a field

On a purely inseparable extension of a normal extension of a field By ROBERT GILMER and BRIAN WESSELINK It is well known that normality of field extensions is not transitive - that is, if F is a subfield of K and if K is a subfield of L, then L/F need not be normal if L/K and K/F are normal. It is true, however, that L/F is normal if L/K is normal and K/F is purely inseparable. It is tempting to conjecture that L/F is normal if L/K is purely inseparable and K/F is normal, and indeed, M. SWEEDLER in [4], Corollary 8, and [5], Corollary 7, asserts that this conjecture is correct. We examine in this paper the question of normality of L/F, where L/K is purely inseparable and K/F is normal. We prove (Theorem 4) that L/F need not be normal. In the case where [L : K] is finite, we determine in Theorem 2 necessary and sufficient conditions in order that L/F be normal; we also investigate related problems, such as that of determining if a field F admits a pair K, L of extension fields with K C_ L, K/F normal, L/K purely inseparable, and L/F not normal (see Theorem 3). All fields are assumed to be http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

On a purely inseparable extension of a normal extension of a field

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02993587
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Abstract

By ROBERT GILMER and BRIAN WESSELINK It is well known that normality of field extensions is not transitive - that is, if F is a subfield of K and if K is a subfield of L, then L/F need not be normal if L/K and K/F are normal. It is true, however, that L/F is normal if L/K is normal and K/F is purely inseparable. It is tempting to conjecture that L/F is normal if L/K is purely inseparable and K/F is normal, and indeed, M. SWEEDLER in [4], Corollary 8, and [5], Corollary 7, asserts that this conjecture is correct. We examine in this paper the question of normality of L/F, where L/K is purely inseparable and K/F is normal. We prove (Theorem 4) that L/F need not be normal. In the case where [L : K] is finite, we determine in Theorem 2 necessary and sufficient conditions in order that L/F be normal; we also investigate related problems, such as that of determining if a field F admits a pair K, L of extension fields with K C_ L, K/F normal, L/K purely inseparable, and L/F not normal (see Theorem 3). All fields are assumed to be

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Nov 18, 2008

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