Mixed degree number field computations

TL;DR

This paper introduces a method for computing complete lists of number fields with specific Galois groups, demonstrated by finding minimal discriminant octic fields with certain Galois groups.

Contribution

The paper presents a novel computational approach for enumerating number fields with particular Galois groups, especially when these groups appear in smaller degree cases.

Findings

Identified 25 octic fields with Galois group PSL_2(7) and minimal discriminant.

Determined the octic field with Galois group 2^3:GL_3(2) of smallest discriminant.

Developed a method applicable to other Galois group computations.

Abstract

We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the twenty-five octic fields with Galois group $\textrm{PSL}_2(7)$ and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group $2^3{:}\textrm{GL}_3(2)$ of smallest absolute discriminant.