Pro-$\ell$-by-cyclotomic and tamely ramified variants of the Neukirch-Uchida Theorem

We prove a generalization of the Neukirch-Uchida Theorem. In particular, we show that the isomorphism type of a number field $K$ can be recovered from the maximal pro-$\ell$-by-cyclotomic quotient of its absolute Galois group $G_{\overline{K}/K}$. This should be contrasted with the previous result that the isomorphism type cannot, in general, be recovered from the maximal pronilpotent quotient. We also show that the isomorphism type can be recovered from the maximal tamely ramified quotient.