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.
Categories: math.NT