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

Shubham Jaiswal, P Vanchinathan

TL;DR

The paper investigates how splitting fields of irreducible polynomials can be generated by subsets of roots, introducing minimal generating sets, root clusters, and cluster towers, and showing that minimal generating sets can have varying cardinalities unlike the linear-algebra setting. It proves a precise equivalence between minimal generating sets and cluster towers, and constructs explicit families, notably for $f(x)=x^n-c$ with odd composite $n$, that realize multiple permissible cardinalities for minimal generating sets and corresponding cluster towers. It then analyzes the influence of Galois-group transitivity on these structures, establishing lower bounds and enumerating examples for which all minimal generating sets have a fixed cardinality, including sharply transitive groups and certain inverse Galois realizations. The results illuminate the interplay between field generation by root subsets and group actions, providing a framework to realize prescribed generating-set sizes and tower lengths across a range of polynomials and base fields.

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].

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

TL;DR

The paper investigates how splitting fields of irreducible polynomials can be generated by subsets of roots, introducing minimal generating sets, root clusters, and cluster towers, and showing that minimal generating sets can have varying cardinalities unlike the linear-algebra setting. It proves a precise equivalence between minimal generating sets and cluster towers, and constructs explicit families, notably for with odd composite , that realize multiple permissible cardinalities for minimal generating sets and corresponding cluster towers. It then analyzes the influence of Galois-group transitivity on these structures, establishing lower bounds and enumerating examples for which all minimal generating sets have a fixed cardinality, including sharply transitive groups and certain inverse Galois realizations. The results illuminate the interplay between field generation by root subsets and group actions, providing a framework to realize prescribed generating-set sizes and tower lengths across a range of polynomials and base fields.

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].
Paper Structure (4 sections, 26 theorems, 9 equations)

This paper contains 4 sections, 26 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Minimal Generating Sets of the Splitting Field of a Polynomial
  3. Minimal Generating Sets and Cluster Towers
  4. Multiple transitivity and Minimal generating sets

Key Result

Theorem 1.1

Let $n>2$ be an odd composite number. Fix $\zeta$ to be a primitive $n$-th root of unity in $\bar{\mathbb{Q}}$. Let $c$ be a positive rational number such that $f=x^n-c$ is an irreducible polynomial over $\mathbb{Q}$. Then

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.3.1
  • Proposition 2.4
  • proof
  • ...and 55 more