Combinatorial Techniques in the Galois Theory of $p$-Extensions

TL;DR

This paper explores combinatorial methods to analyze the structure of Galois groups of p-extensions, providing formulas and techniques to count extensions and understand subgroup filtrations, advancing the understanding of Galois group structures.

Contribution

It introduces new combinatorial techniques and formulas for counting Galois p-extensions and computing graded pieces of filtrations in pro-p groups, aiding Galois group characterization.

Findings

Derived formulas for counting p-extensions over specific fields.

Computed dimensions of graded pieces in Zassenhaus filtrations.

Illustrated the role of small quotients in understanding Galois relations.

Abstract

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.