On Galois extensions with prescribed decomposition groups

JOURNAL OF NUMBER THEORY

卷:220

Article

Kim, Kwang-Seob¹  Koenig, Joachim² 

[1] Chosun Univ, Dept Math, Gwangju, South Korea

[2] Korea Natl Univ Educ, Cheongju, South Korea

关键词: Inverse Galois theory;    Locally cyclic extensions;    Local and global fields;    Unramified extensions;    Galois covers;    Specialization;   

DOI  :  10.1016/j.jnt.2020.06.010

来源: Elsevier

【 摘 要 】

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 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. (c) 2020 The Authors. Published by Elsevier Inc.

【 授权许可】

Free