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

Abstract

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.

28 pages, LaTex: Section 5 is extended by adding Proposition 5.5 and Corollaries 5.6, 5.7

References in corpus (4)

• On the residue fields of Henselian valued stable fields
• Demuskin groups, Galois modules, and the elementary type conjecture
• Algebraic extensions of global fields admitting one-dimensional local class field theory
• Class field theory for strictly quasilocal fields with Henselian discrete valuations

Cited by in corpus (6)

• Algebraic extensions of global fields admitting one-dimensional local class field theory
• Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions
• On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions
• On index-exponent relations over Henselian fields with local residue fields
• Henselian valued quasilocal fields with totally indivisible value groups, II
• Norm groups and class fields of formally real quasilocal fields