

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 vadic 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.

