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

TL;DR

This paper provides explicit methods to construct non-abelian extensions of degree p^3 over the rationals, using elements from cyclotomic extensions, and offers concrete polynomial examples for groups of order 27.

Contribution

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

Findings

Sufficient conditions identified for elements to generate non-abelian p^3-extensions over .

Explicit polynomials for non-abelian groups of order 27 over  are constructed.

Method extends explicit class field theory to non-abelian p^3-extensions.

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.