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Valuations on Division Rings Read chapter Read first chapter

2015 | OriginalPaper | Chapter

5. Graded and Valued Field Extensions

Authors : Jean-Pierre Tignol, Adrian R. Wadsworth

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

Publisher: Springer International Publishing

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Abstract

We pursue in this chapter the investigation of valuations through the graded structures associated to the valuation filtration. The graded field of a valued field is an enhanced version of the residue field, inasmuch as it encapsulates information about the value group in addition to the residue field. It thus captures much of the structure of the field, particularly in the Henselian case. This point is made clear in §5.2, where we show that—when the ramification is tame—Galois groups and their inertia subgroups of Galois extensions of valued fields can be determined from the corresponding extension of graded fields. Henselian fields are shown to satisfy a tame lifting property from graded field extensions, generalizing the inertial lifting property. In §5.1, we lay the groundwork for the subsequent developments by an independent study of graded fields, their algebraic extensions and their Galois theory.

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previous chapter Existence and Fundamental Properties of Gauges
next chapter Brauer Groups
Appendix
Available only for authorised users
Footnotes
1
We could use the right regular representation of S instead of the left in defining these functions. For all the cases considered in this book, either representation gives the same trace, norm, and characteristic polynomial. But this is not true in general, e.g. when S is a ring of triangular matrices over R.
 
2
The graded field extension K/F is tame since the field extension q(K)/q(F) is Galois, hence separable. Therefore, L is also the inertial closure of F in K.
 
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Metadata
Title
Graded and Valued Field Extensions
Authors
Jean-Pierre Tignol
Adrian R. Wadsworth
Copyright Year
2015
Book
Value Functions on Simple Algebras, and Associated Graded Rings

Print ISBN: 978-3-319-16359-8

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

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-16360-4_5

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