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

Let $p$ be an odd prime. Let $F\mathrm{/}k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^3$-extensions over $k$, are induced by certain elements in ${F({\mu }_p)}^{\ast }$. In this paper we let $k=\text{QQ}$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\text{QQ}$ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.