Let $E$ be an algebraic extension of a global field $E_{0}$ with a nontrivialBrauer group Br$(E)$, and let $P(E)$ be the set of those prime numbers $p$, forwhich $E$ does not equal its maximal $p$-extension $E(p)$. This paper showsthat $E$ admits one-dimensional local class field theory if and only if thereexists a system $V(E) = \{v(p)\colon \ p \in P(E)\}$ of (nontrivial) absolutevalues, such that $E(p) \otimes_{E} E_{v(p)}$ is a field, where $E_{v(p)}$ isthe 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 ofBr$(E)$. 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 onecan 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 $p \in P$.
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机译：设$ E $为具有非平凡的Brauer群Br $（E）$的全局字段$ E_ {0} $的代数扩展，令$ P（E）$为素数$ p $的集合，为此， E $不等于其最大$ p $扩展名$ E（p）$。本文表明，当且仅当存在（非平凡的）P（E）\} $中的系统$ V（E）= \ {v（p）\ colon \ p \ in $（E）时，$ E $才接受一维局部类场论绝对值，例如$ E（p）\ otimes_ {E} E_ {v（p）} $是一个字段，其中$ E_ {v（p）} $是相对于$ v（p）的$ E $的完成$。发生这种情况时，我们用$ V（E）$来确定$ E $的有限扩展的范数组以及Br $（E）$的结构。还证明了，如果$ P $是素数的非空集合，而$ \ {w（p）\ colon \ p \ in P \} $是一个绝对值$ E_ {0} $的系统，则可以用这样的理论在$ E_ {0} $上找到字段$ K $代数，使得$ P（K）= P $且元素$ \ kappa（p）\ inV（K）$扩展$ w（p ）$，每个P $中的$ p \。
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