[Paper Review] Relations in the maximal pro-$p$ quotients of absolute Galois groups

Ján Mináč, Michael Rogelstad+1 more|arXiv (Cornell University)|Aug 6, 2018

Algebraic Geometry and Number Theory17 references5 citations

TL;DR

This paper establishes that certain relations in the maximal pro-$p$ quotient of absolute Galois groups—specifically those with a single $p$-th power and commutator terms in a non-symmetric configuration—cannot arise from fields containing a primitive $p$-th root of unity. Using Kummer theory and Galois cohomology, the authors construct explicit small Galois extensions to show that such relations violate the structure constraints imposed by the presence of roots of unity, generalizing classical results of Demushkin and Labute beyond local fields.

ABSTRACT

We observe that some basic but fundamental constructions in Galois theory can be used to obtain some interesting restrictions on the structure of Galois groups of maximal $p$-extensions of fields containing a primitive $p$th root of unity. This is an extension of some significant ideas of Demushkin, Labute and Serre from local fields to all fields containing a primitive $p$th root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.

Motivation & Objective

• To determine which relations can define the maximal pro-$p$ quotient $G_F(p)$ of the absolute Galois group of a field $F$ containing a primitive $p$-th root of unity.
• To extend classical results from local fields (e.g., Demushkin and Labute) to arbitrary fields with a primitive $p$-th root of unity.
• To identify obstructions to the realizability of specific pro-$p$ group presentations as $G_F(p)$ using explicit Galois extensions.
• To clarify the role of $p$-th powers and commutators in defining relations for $G_F(p)$, particularly when the $p$-th power term is not symmetrically placed.

Proposed method

• Constructing simple Galois extensions $F(a,m) = F( oot{p^m} o{a}, \zeta_{p^m})$ for $a \in F^\times$ and $m \in \mathbb{N}$ to test the validity of proposed relations in $G_F(p)$.
• Using Kummer theory to relate the structure of $F^\times / (F^\times)^{p^k}$ to the abelianization $G_F(p)^{ab}$, especially when $\zeta_{p^k} \in F^\times$.
• Applying Galois cohomology and Massey product techniques to detect obstructions in the realizability of certain group presentations as $G_F(p)$.
• Analyzing the torsion and free parts of $G_F(p)^{ab}$ via Pontrjagin duality and the structure of pro-$p$ abelian groups.
• Using the existence of $\zeta_{p^k}$ in $F$ to force the $p$-power torsion in $G_F(p)^{ab}$ to start at $p^k$, restricting possible relation shapes.
• Proving that if $\zeta_{p^k} \in F^\times$, then $G_F(p)^{ab}$ decomposes as a product of $\mathbb{Z}_p$-factors and $p$-torsion of exponent at least $p^k$.

Experimental results

Research questions

• RQ1 Can the relation $r = x_1^{p^s}[x_2,x_3]\cdots[x_{n-1},x_n]$ with $n$ odd and $s \in \mathbb{N}$ be realized as a defining relation in $G_F(p)$ for a field $F$ containing a primitive $p$-th root of unity?
• RQ2 What structural constraints do the presence of $\zeta_{p^k}$ in $F$ impose on the abelianization $G_F(p)^{ab}$, particularly on the exponents of torsion elements?
• RQ3 Why does the shape of the relation $r = x_1^{p^s}[x_1,x_2]\cdots[x_{n-1},x_n]$ (symmetric $p$-power and commutators) allow realizability as $G_F(p)$, while $r = x_1^{p^s}[x_2,x_3]\cdots[x_{n-1},x_n]$ (asymmetric) does not?
• RQ4 How do explicit Galois extensions $F(a,m)$ serve as obstructions to realizing certain relations in $G_F(p)$?
• RQ5 To what extent can techniques from Kummer theory and cohomology be used to distinguish between realizable and non-realizable pro-$p$ group presentations for $G_F(p)$?

Key findings

• The group $G = S / \langle r \rangle$, where $S$ is a free pro-$p$ group on $n$ generators ($n$ odd) and $r = x_1^{p^s}[x_2,x_3]\cdots[x_{n-1},x_n]$ with $s \in \mathbb{N}$, cannot be isomorphic to $G_F(p)$ for any field $F$ containing a primitive $p$-th root of unity.
• When $\zeta_{p^k} \in F^\times$, the torsion part of $G_F(p)^{ab}$ has exponent at least $p^k$, which restricts the possible shapes of defining relations in $G_F(p)$.
• The abelianization $G_F(p)^{ab}$ decomposes as $\prod_J \mathbb{Z}_p \times \prod_{i=1}^l \prod_{m(i)} \mathbb{Z}/p^{s_i}\mathbb{Z}$ with $s_i \geq k$ if $\zeta_{p^k} \in F^\times$, and $l=0$ if $\zeta_{p^k} \in F^\times$ for all $k \geq 1$.
• The construction of the extension $F(a,m) = F(\sqrt[p^m]{a}, \zeta_{p^m})$ provides a concrete obstruction: if a proposed relation $r$ does not vanish in the Galois group of such an extension, it cannot be valid in $G_F(p)$.
• The difference between symmetric and asymmetric placement of the $p$-th power term in the relation—e.g., $x_1^{p^s}[x_1,x_2]\cdots$ vs. $x_1^{p^s}[x_2,x_3]\cdots$—leads to fundamentally different realizability conditions.
• The paper generalizes Labute’s Proposition 6 to infinite pro-$p$ groups, showing that the obstruction mechanism based on $p$-power and commutator structures is robust beyond finite or local settings.