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

TL;DR

This paper provides explicit methods to construct non-abelian extensions of degree p^3 over the rationals, using elements in cyclotomic extensions, and offers polynomial realizations for groups of order 27 with minimal ramification.

Contribution

It introduces explicit criteria for constructing non-abelian p^3-extensions over , specifically over =, and constructs polynomials for groups of order 27 with minimal ramification.

Findings

Explicit criteria for suitable elements in (}_p)^* for constructions.

Polynomial realizations of non-abelian groups of order 27 over .

Construction of extensions with exactly two ramified primes, avoiding Scholz conditions.

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 ${F(\mu_{p})}^{*}$. 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. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.