## Osaka Journal of Mathematics

### Construction of unramified extensions with a prescribed Galois group

Kwang-Seob Kim

#### Abstract

In this article, we shall prove that for any finite solvable group $G$, there exist infinitely many abelian extensions $K/\mathbb{Q}$ and Galois extensions $M/\mathbb{Q}$ such that the Galois group $\Gal(M/K)$ is isomorphic to $G$ and $M/K$ is unramified. The difference between our result and [3, 4, 6, 7, 13] is that we have a base field $K$ which is not only Galois over $\mathbb{Q}$, but also has very small degree compared to their results. We will also get another proof of Nomura's work [9], which gives us a base field of smaller degree than Nomura's. Finally for a given finite nonabelian simple group $G$, we will show there exists an unramified extension $M/K'$ such that the Galois group is isomorphic to $G$ and $K'$ has relatively small degree.

#### Article information

Source
Osaka J. Math., Volume 52, Number 4 (2015), 1039-1051.

Dates
First available in Project Euclid: 18 November 2015

https://projecteuclid.org/euclid.ojm/1447856031

Mathematical Reviews number (MathSciNet)
MR3426627

Zentralblatt MATH identifier
1335.12003

Subjects
Primary: 12F12: Inverse Galois theory
Secondary: 11R29: Class numbers, class groups, discriminants

#### Citation

Kim, Kwang-Seob. Construction of unramified extensions with a prescribed Galois group. Osaka J. Math. 52 (2015), no. 4, 1039--1051. https://projecteuclid.org/euclid.ojm/1447856031

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