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We study the inverse Galois problem with restricted ramifications. Let p  and q  be distinct odd primes such that p≡1modq . 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 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

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