论文标题
在尺寸的字段上是全球或本地字段的Galois扩展
On Fields of dimension one that are Galois extensions of a global or local field
论文作者
论文摘要
让$ k $是全球或本地领域，$ e/k $ agalois扩展名，br $（e）$ $ e $的brauer集团。本文表明，如果$ k $是本地领域，$ v $是其自然离散估值，$ v'$是$ e $延长$ v $的估值，而$ q $是残留场$ \ widehat e $ of $ of $（e，v'）$的特征$ p $ - $ \ widehat k $的扩展，对于每个prime $ p \ neq q $; $ \ widehat e $是一个代数封闭的字段，以防Value Group $ V'（e）$是$ q $ - 不可分割的。当$ k $是一个全球字段时，它以br $（e）= \ {0 \} $为特征，该字段$ e $位于$ k $的Tame Abelian扩展名中。我们还提供了一个标准，即在后一种情况下，对于任何整数$ n \ ge 2 $，都存在$ n $ variate $ e $ - $ n $的形式，违反了Hasse原则。
Let $K$ be a global or local field, $E/K$ a Galois extension, and Br$(E)$ the Brauer group of $E$. This paper shows that if $K$ is a local field, $v$ is its natural discrete valuation, $v'$ is the valuation of $E$ extending $v$, and $q$ is the characteristic of the residue field $\widehat E$ of $(E, v')$, then Br$(E) = \{0\}$ if and only if the following conditions hold: $\widehat E$ contains as a subfield the maximal $p$-extension of $\widehat K$, for each prime $p \neq q$; $\widehat E$ is an algebraically closed field in case the value group $v'(E)$ is $q$-indivisible. When $K$ is a global field, it characterizes the fields $E$ with Br$(E) = \{0\}$, which lie in the class of tame abelian extensions of $K$. We also give a criterion that, in the latter case, for any integer $n \ge 2$, there exists an $n$-variate $E$-form of degree $n$, which violates the Hasse principle.
