Some Remarks on the Existence of Certain Unramified $p$-extensions

Abstract

We study the inverse Galois problem with restricted ramifications. Let $p$ and $q$ be distinct odd primes. Let $E$ be a non-abelian $p$-group of order $p^3$, and let $k$ be a cyclic extension over ${\mathbf Q}$ of degree $q$. In this paper, we study the existence of unramified extensions over $k$ with the Galois group $E$.