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

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.

arXiv comment8 pages, to appear on the Annales mathématiques du Québec

References in corpus (2)

• Bloch-Kato pro-p groups and locally powerful groups
• The Bloch-Kato Conjecture and Galois Theory

Cited by in corpus (2)

• On pro-$p$ groups with quadratic cohomology
• One relator maximal pro-p Galois groups and the Koszulity conjectures