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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 ofunity. Denote the absolute Galois group of $K$ by $\operatorname{Gal}(K)$, andthe mod-$n$ central-descending series of $\operatorname{Gal}(K)$ by$\operatorname{Gal}(K)^{(i)}$. Recall that Kummer theory, together with ourchoice 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, inthis 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 groupof $(\mathbb Z/n)^2$-valued functions on $K\smallsetminus\{0,1\}$. Weexplicitly determine the functions associated to the image of commutators and$n$th powers of elements of $\operatorname{Gal}(K)$ under this embedding. Wethen apply this theory to prove some new results concerning relations betweenelements in abelian-by-central Galois groups.

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