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

Abstract

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 ${F({\mu }_p)}^{\ast }$. 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.

10 pages. Keywords: Constructive Galois Theory; Heisenberg group, Explicit Embedding problem. Corrections: we revised the proof of Theorem 3.1