Notes on the existence of unramified non-abelian $p$-extensions over cyclic fields

Abstract

We study the inverse Galois problem with restricted ramifications. Let $p$ and $q$ be distinct odd primes such that $p\equiv 1 \bmod q$. Let $E(p^{3})$ be the non-abelian group of order $p^{3}$ such that the exponent is equal to $p$, 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 $E(p^{3})$. We also give some numerical examples computed with PARI.