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

We study the inverse Galois problem with restricted ramifications. Let p and q be distinct odd primes such that p ≡ 1 mod q. Let E(p³) be the non-abelian group of order p³ such that the exponent is equal to p, and let k be a cyclic extension over Q of degree q. In this paper, we study the existence of unramified extensions over k with the Galois group E(p³). We also give some numerical examples computed with PARI. [ABSTRACT FROM AUTHOR]