A Presentation of the Absolute Galois Group of $\mathbb{Q}_2$

FrontierMath: Open Problems lists this problem with notability “Solid result”, field “Number theory” and type “Construction - Finite”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.

Verifiable task — Epoch prompt, verbatim

Find a presentation for the absolute Galois group of the field of 2-adic numbers.  It may be helpful to know that the maximal pro-2 quotient of the absolute Galois group can be presented with three generators x,y,z and one relation, x^2*y^3*y^z=1.  The theory of Demuskin groups may also be useful.

You should give your output as two or three sections, separated by blank lines: variables, definitions (optional) and relations.

The variables section should consist of a comma separated list of variable names on one line corresponding to the generators of the presentation.  The first and second generators (say $\sigma$ and $\tau$ respectively) are distinguished, and must satisfy the implicit relation $\tau^\sigma = \tau^2$ (which should not be included in the \texttt{relations} section).  All later generators should be contained within the $2$-core of the group (the intersection of all $2$-Sylow subgroups).

In the definitions section, you may define additional variables in terms of the generators and variables defined earlier within the definitions section.  Each line should take the form "var = expr", where expr is an expression built using group operations.

In the relations section, you should give the relations as expressions in terms of variables defined in the previous two sections.  If there are multiple relations, give one on each line.

Expressions in profinite groups may involve exponents lying in $\hat{\Z}$.  To specify such a constant, you should provide its reduction modulo $85667662080 = 2^8 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23$.

Example format for the solution (this is a presentation for p=3):

sigma,tau,x0,x1

tau2 = tau^3011753745
sigma2 = sigma^3011753745
rho = tau2^4
angle = (x0*tau*x0^-1*tau)^2974571600
u = (x1*(rho^2)*x1*(rho^2))^2974571600
v = sigma2*tau2
w = (u*(v^2)*u^-1*(v^2))^2974571600
y1 = x1^rho*(u^v)*(w^(sigma2*tau2^2))*(w^(tau2^2))
comm = x1^-1*y1^-1*x1*y1

(x0^sigma)^-1*angle*x1^3*comm

Epoch marks this problem solved (AI) on 2026-06-09. This is Epoch’s classification; the solution has not been independently checked for this record.

Epoch solution update

First solved by: Claude Fable 5

This problem has been solved! See this write-up, by David Roe and David Turturean, for details.