# Galois groups and cohomological functors

@article{Efrat2011GaloisGA, title={Galois groups and cohomological functors}, author={Ido Efrat and J{\'a}n Min{\'a}{\vc}}, journal={arXiv: Number Theory}, year={2011} }

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring $H^*(G_F,\mathbb{Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E…

## 28 Citations

Detecting Fast solvability of equations via small powerful Galois groups

- Mathematics
- 2013

Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the…

Abelian-by-Central Galois groups of fields I: a formal description

- Mathematics
- 2013

Let $K$ be a field whose characteristic is prime to a fixed integer $n$ with $\mu_n \subset K$, and choose $\omega \in \mu_n$ a primitive $n$th root of unity. Denote the absolute Galois group of $K$…

3-fold Massey products in Galois cohomology -- The non-prime case

- Mathematics
- 2020

For $m\geq2$, let $F$ be a field of characteristic prime to $m$ and containing the roots of unity of order $m$, and let $G_F$ be its absolute Galois group. We show that the 3-fold Massey products…

Counting Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions using Massey products

- Mathematics
- 2014

We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed…

Reconstructing function fields from rational quotients of mod-$$\ell $$ℓ Galois groups

- Mathematics
- 2014

In this paper, we develop the main step in the global theory for the mod-$$\ell $$ℓ analogue of Bogomolov’s program in birational anabelian geometry for higher-dimensional function fields over…

Combinatorial Techniques in the Galois Theory of $p$-Extensions

- Mathematics
- 2015

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of…

The lower p-central series of a free profinite group and the Shuffle algebra

- MathematicsJournal of Pure and Applied Algebra
- 2020

Triple Massey products and absolute Galois groups

- Mathematics
- 2014

Let $p$ be a prime number, $F$ a field containing a root of unity of order $p$, and $G_F$ the absolute Galois group. Extending results of Hopkins, Wickelgren, Minac and Tan, we prove that the triple…

THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2021

For a prime number p and a free profinite group S on the basis X, let
$S_{\left (n,p\right )}$
,
$n=1,2,\dotsc ,$
be the p-Zassenhaus filtration of S. For
$p>n$
,…

Small Galois groups that encode valuations

- Mathematics
- 2011

Let $p$ be a prime number and let $F$ be a field containing a root of unity of order $p$. We prove that a certain very small canonical Galois group $(G_F)_{[3]}$ over $F$ encodes the valuations on…

## References

SHOWING 1-10 OF 25 REFERENCES

Quotients of absolute Galois groups which determine the entire Galois cohomology

- Mathematics
- 2009

For a prime power q = pd and a field F containing a root of unity of order q we show that the Galois cohomology ring $${H^*(G_F,\mathbb{Z}/q)}$$ is determined by a quotient $${G_F^{[3]}}$$ of the…

On the descending central sequence of absolute Galois groups

- Mathematics
- 2008

Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group $G_F$ of $F$.…

Small Galois groups that encode valuations

- Mathematics
- 2011

Let $p$ be a prime number and let $F$ be a field containing a root of unity of order $p$. We prove that a certain very small canonical Galois group $(G_F)_{[3]}$ over $F$ encodes the valuations on…

The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields

- Mathematics
- 2013

Galois module structure of (ℓn)th classes of fields

- Mathematics
- 2012

In this paper, we use the Merkurjev–Suslin theorem to determine the structure of arithmetically significant Galois modules that arise from Kummer theory. Let K be a field of characteristic different…

Homology and Cohomology of Profinite Groups

- Mathematics
- 2010

In this section we introduce some terminology and sketch some general homological results. We shall state the concepts and results for general abelian categories to avoid repetitions, but we are…

On the birational anabelian program initiated by Bogomolov I

- Mathematics
- 2012

Recall that a program initiated by Bogomolov in 1990 aims at reconstructing function fields K|k with td(K|k)>1 and k algebraically closed from the maximal pro-ℓ abelian-by-central Galois group $\Pi…

K-cohomology of Severi-Brauer Varieties and the norm residue homomorphism

- Mathematics
- 1983

The basic purpose of this paper is to prove bijectivity of the norm residue homomorphism for any field of characteristic prime to . In particular, if , then any central simple algebra of exponent is…

Cohomology of number fields

- Mathematics
- 2000

Part I Algebraic Theory: Cohomology of Profinite Groups.- Some Homological Algebra.- Duality Properties of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa Modules.- Part II Arithmetic…