Abelian-by-Central Galois groups of fields I: a formal description

TL;DR

This paper develops a formal description of abelian-by-central Galois groups of fields, constructing explicit embeddings and analyzing relations using advanced algebraic tools like the Merkurjev-Suslin theorem.

Contribution

It introduces a functorial, explicit embedding of certain Galois group quotients into function groups, extending Kummer theory with new algebraic insights.

Findings

Constructed explicit embeddings of Galois group quotients

Determined functions associated with commutators and powers

Proved new relations in abelian-by-central Galois groups

Abstract

Let $K$ be a field whose characteristic is prime to a fixed integer $n$ with $\mu_n \subset K$, and choose $\omega \in \mu_n$ a primitive $n$th root of unity. Denote the absolute Galois group of $K$ by $\operatorname{Gal}(K)$, and the mod-$n$ central-descending series of $\operatorname{Gal}(K)$ by $\operatorname{Gal}(K)^{(i)}$. Recall that Kummer theory, together with our choice of $\omega$, provides a functorial isomorphism between $\operatorname{Gal}(K)/\operatorname{Gal}(K)^{(2)}$ and $\operatorname{Hom}(K^\times,\mathbb{Z}/n)$. Analogously to Kummer theory, in this note we use the Merkurjev-Suslin theorem to construct a continuous, functorial and explicit embedding $\operatorname{Gal}(K)^{(2)}/\operatorname{Gal}(K)^{(3)} \hookrightarrow \operatorname{Fun}(K\smallsetminus\{0,1\},(\mathbb Z/n)^2)$, where $\operatorname{Fun}(K\smallsetminus\{0,1\},(\mathbb Z/n)^2)$ denotes the group of $(\mathbb Z/n)^2$-valued functions on $K\smallsetminus\{0,1\}$. We explicitly determine the functions associated to the image of commutators and $n$th powers of elements of $\operatorname{Gal}(K)$ under this embedding. We then apply this theory to prove some new results concerning relations between elements in abelian-by-central Galois groups.