  Let $p$ be an odd prime. Let $F/k$ 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
${F(\mu_{p})}^{*}$. In this paper we let $k=\QQ$ and present sufficient
conditions for these elements to be suitable for the constructions. Polynomials
for the non-abelian groups of order 27 over $\QQ$ are constructed.
