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

Abstract:

Let $K$ be a field whose characteristic is prime to a fixed positive integer $n$ such that $\mu _n \subset K$, and choose $\omega \in \mu _n$ as 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.