A famous theorem of Neukirch and Uchida states that the isomorphy type of a number field is functorially encoded in the isomorphy type of its absolute (pro-solvable) Galois group. During the summer 2017, together with Akio Tamagawa, we proved the following theorems.

 

The isomorphy type of number field is determined by the isomorphy type of its maximal 3-step solvable Galois group. Further, the isomorphy type of the maximal m-step solvable extension of a number field is determined (resp. functorially) by the isomorphy type of its maximal (m+4)-step (resp. (m+5)-step) solvable Galois group. 

 

This is a substantial sharpening of the Neukirch and Uchida theorem. In my talk I will review these theorems, give some hints about the ideas of proofs, discuss relations to other works, and discuss some open questions as well as the impact these theorems might have in the future.