We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realizations with only cyclic decomposition groups and with nonsolvable Galois group. We also investigate the analogous problem over global function fields.

中文翻译：

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具有指定分解群的伽罗瓦扩展

我们研究了局部条件下的逆伽罗瓦问题。特别地，我们询问是否每个有限群都作为 $\mathbb{Q}$ 的伽罗瓦扩展的伽罗瓦群出现，其所有分解群都是循环的（即阿贝尔）。由于 Shafarevich 对这些群的逆伽罗瓦问题的解，所有可解群的这一性质都是已知的。然而，它对不可解群是完全开放的。在本文中，我们通过函数域扩展的特化提供了解决这些问题的一般标准，特别是给出了只有循环分解群和不可解伽罗瓦群的伽罗瓦实现的第一个无限族。我们还研究了全局函数域上的类似问题。