Lightly ramified number fields with Galois group $S.M_{12}.A$

TL;DR

This paper constructs specific number fields with Galois groups related to M12, exhibiting minimal ramification, including a unique case ramified only at prime 11, advancing understanding of ramification in Galois extensions.

Contribution

It introduces new explicit constructions of number fields with complex Galois groups and minimal ramification, including a rare example ramified only at prime 11.

Findings

Constructed number fields with Galois groups M12, M12.2, 2.M12, 2.M12.2

Identified a field ramified only at prime 11

Demonstrated methods to achieve light ramification in Galois extensions

Abstract

We specialize various three-point covers to find number fields with Galois group $M_{12}$, $M_{12}.2$, $2.M_{12}$, or $2.M_{12}.2$ and light ramification in various senses. One of our $2.M_{12}.2$ fields has the unusual property that it is ramified only at the single prime $11$.