The Index Problem for Subgroup Intersections

TL;DR

This paper investigates the realization of triples of positive integers as degrees of number fields and their compositum, using group-theoretic reformulation to simplify and extend previous field-theoretic results.

Contribution

It reformulates the index problem for subgroup intersections in the language of groups, providing a more general and simplified approach to understanding realizable triples.

Findings

Provides a group-theoretic framework for the index problem.

Generalizes previous results from field theory to group theory.

Simplifies the characterization of realizable triples.

Abstract

In previous papers, Drungilas et al. study the problem of which triples of positive integers $(a, b, c)$ can be realized as $([E: \mathbb{Q}], [F: \mathbb{Q}], [EF: \mathbb{Q}])$, where $E$ and $F$ are number fields, using techniques from field theory. We shall study this problem rephrased in the language of groups using the Galois correspondence to simplify and generalize their results.