About the problem
The absolute Galois group of a field \(K\) is the projective limit of all of the finite Galois groups \(\operatorname{Gal}(E/K)\). It packages together the information about all finite extensions, and studying the Galois group of the rational field \(\mathbb{Q}\) is a central problem in algebraic number theory. One method for approaching this group is to study the analogous Galois groups of p-adic fields \(\mathbb{Q}_p\). In this case, there is an explicit presentation of \(\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\) for \(p>2\). For \(p=2\), we have a description for the absolute Galois group of some extensions of \(\mathbb{Q}_2\), but not for \(\mathbb{Q}_2\) itself. Finding such a description would fill in a gap in our explicit understanding of Galois groups, and would have consequences in terms of counting p-adic fields with a given Galois group.
Progress Update (2026-07-06): Multiple AI systems have now generated solutions that are accepted by the verifier for this problem. The first solution we were made aware of was generated by Claude Fable 5, elicited by problem contributor David Roe on 2026-06-09. Then, on 2026-06-24, David Turturean wrote in with a solution generated by GPT-5.5 Pro running in a harness he designed. We believe that these solutions are likely to be correct, though it remains to check that the AI systems can actually prove this. In general FrontierMath: Open Problems verifiers provide strong evidence that an AI system has made meaningful progress on a problem, but for some problems, including this one, the evidence they provide is short of a full proof. In any case, since Fable was unavailable, Roe and Turturean began working to understand whether GPT-5.5 Pro could produce a full proof for its solution. We will provide an update here when they reach a conclusion.
Warm-up: We ask for a presentation of the absolute Galois group of \(\mathbb{Q}_3\), which is known.
AI prompts
Warm-up
Find a presentation for the absolute Galois group of the field of 3-adic numbers. It may be helpful to know that the maximal pro-3 quotient of the absolute Galois group is a free group on 2 generators, and that for every finite tamely ramified extension $K/\mathbb{Q}_3$ the maximal pro-3 quotient of $\Gal(\overline{K}/K)$ is a Demuskin group.
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 $\tau^\sigma = \tau^3$ (which should not be included in the \texttt{relations} section). All later generators should be contained within the $3$-core of the group (the intersection of all $3$-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 $5354228880 = 2^4 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23$.
Example format for the solution (this is a presentation for the maximal pro-2 quotient of the 2-adic absolute Galois group
z,tau,x,y
x^2*y^3*y^z Full problem
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 Mathematician survey
The author assessed the problem as follows.
a majority of those working on a specialized topic (≈10)
2–4
1–3 months
a solid result in a subfield
in a standard specialty journal
somewhat likely: an especially innovative solution could, but is hardly guaranteed
95-99%