The Kummerian Property and Maximal Pro-$p$ Galois Groups

Abstract

For a prime number $p$, we give a new restriction on pro-$p$ groups $G$ which are realizable as the maximal pro-$p$ Galois group $G_F(p)$ for a field $F$ containing a root of unity of order $p$. This restriction arises from Kummer Theory and the structure of the maximal $p$-radical extension of $F$. We study it in the abstract context of pro-$p$ groups $G$ with a continuous homomorphism $\theta :G\to 1+p{\mathbb{Z}}_p$, and characterize it cohomologically, and in terms of 1-cocycles on $G$. This is used to produce new examples of pro-$p$ groups which do not occur as maximal pro-$p$ Galois groups of fields as above.

Final revised version. To appear in the Journal of Algebra

References in corpus (5)

• Koszul property and Bogomolov's conjecture
• Bloch-Kato pro-p groups and locally powerful groups
• Triple Massey products in Galois cohomology
• Profinite groups with a cyclotomic $p$-orientation
• Relations in the maximal pro-$p$ quotients of absolute Galois groups

Cited by in corpus (15)

• On pro-$p$ groups with quadratic cohomology
• Profinite groups with a cyclotomic $p$-orientation
• Groups of p-absolute Galois type that are not absolute Galois groups
• Enhanced Koszul properties in Galois cohomology
• Galois-theoretic features for 1-smooth pro-$p$ groups
• Right-angled Artin groups and enhanced Koszul properties
• Oriented pro-$\ell$ groups with the Bogomolov-Positselski property
• Right-angled Artin pro-$p$ groups
• Massey products in Galois cohomology and the Elementary Type Conjecture
• Two families of pro-p groups that are not absolute Galois groups
• Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups
• One relator maximal pro-p Galois groups and the Koszulity conjectures
• Chasing maximal pro-p Galois groups via 1-cyclotomicity
• Frattini-injectivity and Maximal pro-$p$ Galois groups
• Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups