Abelian-by-central Galois groups of fields I: A formal description

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.