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It is easily seen that if | · | is non-archimedean and x, y ∈ K with |x| < |y|, then |x + y| = max(|x|, |y|) = |y|. Two absolute values on a field are said to be equivalent if they define the same topology. | · | is called the trivial absolute value on K if |x| = 1 for all x 6= 0. Example.
This book presents local class field theory from the cohomological point of view, following the method of Hochschild and Artin-Tate. It covers basic facts on local fields, ramification, group cohomology, and local class field theory.
In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology.
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A book that introduces the basic concepts and results of local fields, their extensions and applications. It covers topics such as valuation theory, class field theory, group cohomology, Fontaine theory and more.
Hace 3 días · A field which is complete with respect to a discrete valuation is called a local field if its field of residue classes is finite. The Hasse principle is one of the chief applications of local field theory.
In this lecture we give a brief overview of local class field theory. Recall that a local field is a locally compact field whose topology is induced by a nontrivial absolute value (Def-inition9.1). As we proved in Theorem 9.9, every local field is isomorphic to one of the following: • R orC (archimedean,characteristic0 ...
