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 $V(E) = \{v(p)\colon \ p \in P(E)\}$ 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 $\{w(p)\colon \ p \in P\}$ 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 $p \in P$.