论文标题
建造具有规范规范的Abelian扩展
Constructing abelian extensions with prescribed norms
论文作者
论文摘要
鉴于一个数字字段$ k $，有限的Abelian Group $ g $以及有限的许多元素$α_1，\ ldots，α_t\ in K $，我们构建了Abelian Extensions $ L/K $，使用Galois $ g $ galois $ g $认识到所有元素$α_1，\ ldots，\ ldots，\ ldots，\ ldots，α_t$ as nard aS nord nord y nord $ l l $ l l l l $ l l ld $ l ld $ l l l ld $ l。特别是，这显示了任何给定参数的扩展存在。 我们的方法依赖于阶级领域理论以及泰特（Tate）对圣规范原理的表征的最新表述，这是一种局部全球原则。这些结构足够明确以在计算机上实现，我们用具体的示例来说明它们。
Given a number field $K$, a finite abelian group $G$ and finitely many elements $α_1,\ldots,α_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $α_1,\ldots,α_t$ as norms of elements in $L$. In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate's characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples.
