Construction of unramified extensions with a prescribed Galois group

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.