Algebraic extensions of global fields admitting one-dimensional local class field theory

Abstract

Let $E$ be an algebraic extension of a global field $E_0$ with a nontrivial Brauer group Br$(E)$, and let $P(E)$ be the set of those prime numbers $p$, for which $E$ does not equal its maximal $p$-extension $E(p)$. This paper shows that $E$ admits one-dimensional local class field theory if and only if there exists a system $V(E)=\{v(p):p\in P(E)\}$ of (nontrivial) absolute values, such that $E(p){\otimes }_E{E}_{v(p)}$ is a field, where $E_{v(p)}$ is the completion of $E$ with respect to $v(p)$. When this occurs, we determine by $V(E)$ the norm groups of finite extensions of $E$, and the structure of Br$(E)$. It is also proved that if $P$ is a nonempty set of prime numbers and $\{w(p):p\in P\}$ is a system of absolute values of $E_0$, then one can find a field $K$ algebraic over $E_0$ with such a theory, so that $P(K)=P$ and the element $\kappa (p)\in V(K)$ extends $w(p)$, for each $p\in P$.

23 pages, slightly abridged, a serious mistake in Definition 3, (c), fixed

References in corpus (2)

• Primarily quasilocal fields and 1-dimensional abstract local class field theory
• On the scope of validity of the norm limitation theorem in one-dimensional abstract local class field theory

Cited by in corpus (4)

• On the residue fields of Henselian valued stable fields
• Primarily quasilocal fields and 1-dimensional abstract local class field theory
• Class field theory for strictly quasilocal fields with Henselian discrete valuations
• On the scope of validity of the norm limitation theorem in one-dimensional abstract local class field theory