Finite quotients of Galois pro-$p$ groups and rigid fields

TL;DR

This paper demonstrates that the coincidence of certain finite quotients of a Bloch-Kato pro-$p$ group implies the group is $p$-adic analytic, leading to a Galois-theoretic characterization of $p$-rigid fields.

Contribution

It establishes a new criterion linking finite quotients of Bloch-Kato pro-$p$ groups to their $p$-adic analytic structure, with implications for Galois theory.

Findings

Coincidence of specific finite quotients implies $p$-adic analyticity.

If two canonical extensions of a field coincide, the field is $p$-rigid.

Proof relies solely on group-theoretic methods and properties of Bloch-Kato groups.

Abstract

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.