Primarily quasilocal fields and 1-dimensional abstract local class field theory

Details Primarily quasilocal fields and 1-dimensional abstract local class field theory Primarily quasilocal fields and 1-dimensional abstract local class field theory

2005-06-24

arxiv.org/abs/math/0506515v7

Mathematics

Rings and Algebras

24 pages, LaTex: The name of the title is changed; the contents of Section 4 of previous versions is now presented in Sections

Scientific paper

Let $E$ be a field satisfying the following conditions: (i) the $p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$ is a prime number for which $E$ is properly included in its maximal $p$-extension; (ii) the relative Brauer group Br$(L/E)$ equals the maximal subgroup of Br$(E)$ of exponent $p$, for every cyclic extension $L/E$ of degree $p$. The paper proves that finite abelian extensions of $E$ are uniquely determined by their norm groups and related essentially as in the classical local class field theory. This includes analogues to the fundamental correspondence, the local reciprocity law and the local Hasse symbol.

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