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

TL;DR

This paper generalizes the Neukirch-Uchida Theorem by demonstrating that a number field's isomorphism type can be reconstructed from specific quotients of its Galois group, including pro-$ ext{ extlbrackdbl} ext{ extbrackdbl}$-by-cyclotomic and tamely ramified quotients.

Contribution

It introduces new methods to recover the number field's isomorphism type from maximal pro-$ ext{ extlbrackdbl} ext{ extbrackdbl}$-by-cyclotomic and tamely ramified quotients of the Galois group, extending previous results.

Findings

Number field isomorphism can be recovered from maximal pro-$ ext{ extlbrackdbl} ext{ extbrackdbl}$-by-cyclotomic quotient.

Number field isomorphism can be recovered from maximal tamely ramified quotient.

Contrast with previous results showing limitations of pronilpotent quotients.

Abstract

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.