On Minimal generating sets of splitting field, Cluster towers and Multiple transitivity of Galois groups

TL;DR

This paper explores the properties of minimal generating sets of Galois extensions, revealing their variable sizes, their behavior under multiple transitivity, and their connection to root cluster towers, with specific results for polynomials over the rationals.

Contribution

It introduces a natural notion of minimal generating sets for Galois extensions and analyzes their cardinalities, behavior under transitivity, and relation to root cluster towers, providing new insights into Galois group structures.

Findings

Minimal generating sets can have different cardinalities for certain polynomials.

Existence of polynomials with all minimal generating sets of the same size.

Established connections between minimal generating sets and root cluster towers.

Abstract

A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous concept in linear algebra: they can be of different cardinalities. In fact we establish that for a certain family of polynomials over the rationals, we have minimal generating sets of all cardinalities in a certain range and that these are the only possible cardinalities for minimal generating set for such a polynomial. We also study how minimal generating sets behave under multiple transitivity of the Galois group and consequently prove the existence of polynomials with all minimal generating sets of uniformly same cardinality. We also connect minimal generating sets with the concept of root cluster tower of an irreducible polynomial introduced by the second author and Krithika in [8].