CONSTRUCTION OF UNRAMIFIED EXTENSIONS WITH A PRESCRIBED GALOIS GROUP

Abstract

In this article, we shall prove that for any finite solvable gr oup G, there exist infinitely many abelian extensions K=Q and Galois extensionsM=Q such that the Galois group Gal( M=K ) is isomorphic toG and M=K is unramified. The difference between our result and [3, 4, 6, 7, 13] is that we have a base fiel d K which is not only Galois overQ, but also has very small degree compared to their results. We will also get another proof of Nomura’s work [9], which gives u a base field of smaller degree than Nomura’s. Finally for a given finite nona beli n simple groupG, we will show there exists an unramified extension M=K 0 such that the Galois group is isomorphic toG and K 0 has relatively small degree.