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Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and ofcharacteristic different from $p$. The explicit constructions of thenon-abelian $p^{3}$-extensions over $k$, are induced by certain elements in${F(\mu_{p})}^{*}$. In this paper we let $k=\QQ$ and present sufficientconditions for these elements to be suitable for the constructions. Polynomialsfor the non-abelian groups of order 27 over $\QQ$ are constructed. We describeexplicit realizations of those groups with exactly two ramified primes, withoutconsider Scholz conditions.

Explicit Constructions of the non-Abelian $\mathbf{p^3}$-Extensions Over $\mathbf{\QQ}$