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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Geometric characterization of strongly normal extensions

Author: Jerald J. Kovacic
Journal: Trans. Amer. Math. Soc. 358 (2006), 4135-4157
MSC (2000): Primary 12H05, 12F10; Secondary 14A15, 14L15
Published electronically: April 11, 2006
MathSciNet review: 2219014
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Abstract: This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use ``group chunks'' or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.

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Jerald J. Kovacic
Affiliation: Department of Mathematics, The City College of The City University of New York, New York, New York 10031

PII: S 0002-9947(06)03868-2
Received by editor(s): December 12, 2003
Received by editor(s) in revised form: September 19, 2004
Published electronically: April 11, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.