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Title of Thesis

Algebraic Elements and their Arithmetic in Banach Algebras of Continuous Functions on Galois Group

Author(s)

Sobia Sultana

Institute/University/Department Details
Abdus Salam School of Mathematical Sciences / GC University, Lahore
Session
2009
Subject
Mathematics
Number of Pages
80
Keywords (Extracted from title, table of contents and abstract of thesis)
Algebraic, Elements, Arithmetic, Banach, Algebras, Spectral, norm, Continuous, Functions, Galois, Group, Kransner’s, Lemma

Abstract
Since the Galois absolute group G=Gal (Q/Q) does not act continuously on Q, the usual topology of C cannot be directly used to study the group G (Q is dense in C). in this thesis we consider finite (or infinite) extension L of Q, in Q and its corresponding absolute group GL = Gal (Q/L). On Q we introduce a norm,
x->||x|| L = max {|σ(x) | : σ ε GL }
and call it the L-(or GL -) Spectral norm. Let QL be the completion of Q with respect to ||.||L . Since ||σ (x) ||L = ||x ||L for any σ ε GL , GL acts continuously on Q. We prove many results on the structure of Q L and we connect it with GL itself.
We also introduce the new notion of a v-adic maximal extension L (v) of a valued field (K, v) and we supply with some fundamental results relative to the structure of L (v) . We also connect it with some particular types of spectral norms.
Some other auxiliary results (a strange functional generalization of Kransner’s Lemma, for instance) which are useful by their own are given.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

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67 KB
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1

PRELIMINARIES IN FIELD THEORY AND THE KRULL'S TOPOLOGY ON GALOIS GROUPS 7
706 KB
3 2 A NON-ARCHIMEDEAN GENERALIZATION OF KRASNER'S LEMMA 19
403 KB
4 3 AN ARCHIMEDEAN GENERALIZATION OF KRASNER'S LEMMA 27
241 KB
5 4 THE ALGEBRA C (GK) 32
1,020 KB
6 5 KRASNER'S TYPE LEMMAS 48
167 KB
7 6 V-ADIC MAXIMAL EXTENSIONS, SPECTRAL NORMS AND ABSOLUTE GALOIS GROUPS 51
1,019 KB
8 7 BIBLIOGRAPHY 68
148 KB