Retract rational fields and cyclic Galois extensions |
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Authors: | David J Saltman |
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Institution: | (1) Department of Mathematics, The University of Texas, 78712 Austin, TX, USA |
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Abstract: | In 23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point
was the connection between these problems and Noether’s problem. In 24], a similar sort of study was begun for central simple
algebras, with a connection to the center of generic matrices. In 25], the notion of retract rational field extension was
defined, and a connection with lifting questions was claimed, which was used to complete the results in 23] and 24] about
Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting
problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties
are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract
rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection
with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field.
The author is grateful for NSF support under grant #MCS79-04473. |
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