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2015 | OriginalPaper | Buchkapitel

9. Subfields and Splitting Fields of Division Algebras

verfasst von : Jean-Pierre Tignol, Adrian R. Wadsworth

Erschienen in: Value Functions on Simple Algebras, and Associated Graded Rings

Verlag: Springer International Publishing

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Abstract

In this chapter we apply the machinery developed in previous chapters to analyze the subfields and splitting fields of division algebras over a Henselian field F. In §9.1 we give properties of the splitting fields of tame division algebra D with center F, with particularly strong criteria proved if D is inertial or totally ramified over F. This leads to explicit constructions of several interesting examples of division algebras, including noncyclic division algebras of degree p ² with no maximal subfield of the form \(F(\!\sqrt[p^{2}]{a})\) in Examples 9.15, 9.17, and 9.18; noncyclic p-algebras in Ex. 9.26; noncrossed product algebras including universal division algebras in Th. 9.30 and division algebras over Laurent series over \(\mathbb {Q}\), noncrossed products whose degree exceeds the exponent in Cor. 9.46; and crossed product division algebras with only one Galois group for all maximal subfields Galois over the center in Prop. 9.28[9.28].

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Vorheriges Kapitel Division Algebras over Henselian Fields

Nächstes Kapitel Indecomposable Division Algebras

Nur mit Berechtigung zugänglich

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Titel: Subfields and Splitting Fields of Division Algebras
verfasst von: Jean-Pierre Tignol
Adrian R. Wadsworth
Verlag: Springer International Publishing
Buch: Value Functions on Simple Algebras, and Associated Graded Rings
Print ISBN: 978-3-319-16359-8

Electronic ISBN: 978-3-319-16360-4

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-16360-4_9

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