Title of Thesis

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

Author(s)

Sobia Sultana

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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