Epoch's FrontierMath Benchmark Cracks a 40-Year-Old Math Problem Using Fable 5
Epoch AI announces AI has cracked a 40-year-old open problem in number theory, the second solved on its FrontierMath: Open Problems benchmark.
- Epoch AI announces AI has solved the 2-adic absolute Galois group problem, open since the early 1980s, on its FrontierMath: Open Problems benchmark.
- Two independent solutions were found: one elicited by problem author David Roe using Fable 5, another by David Turturean using GPT-5.5 Pro over a 26-hour session.
- The result was validated against 5,402 finite test groups by an automated verifier, and separately formalized in Lean 4 by both researchers.
- It is the first problem solved in the "Solid Result" tier — judged publishable in a standard specialty journal by consulted mathematicians.
- The full interactive paper is publicly available, including two independent Lean 4 formalizations and a comparison of both AI-found presentations.
- Epoch AI says an expanded Open Problems problem set is coming within the week.
Epoch AI has announced that AI has produced an explicit presentation for the absolute Galois group of the field of 2-adic numbers, a problem that resisted mathematicians since the early 1980s. It is the second problem solved on FrontierMath: Open Problems, Epoch's benchmark of genuinely unsolved research mathematics, and the first in the "Solid Result" category, meaning it carries real weight in its subfield.
What was actually solved
The absolute Galois group of a field is a single algebraic object that encodes all of that field's extension theory at once: every way you can adjoin roots and build larger number systems. For p-adic fields, which are number systems built around a prime p that capture modular arithmetic in a continuous way, explicit presentations of this group were known for all odd primes since the early 1980s, thanks to work by Jannsen and Wingberg. The prime 2 was the stubborn exception.
No explicit presentation of the absolute Galois group of Q₂ had ever been written down: no concrete generators, no concrete relations. The new result fills that gap, giving an explicit presentation in an enriched, "marked" sense that extends Jannsen and Wingberg's framework for odd primes.
Concretely, the absolute Galois group of Q₂ is the profinite group generated by four marked elements σ, τ, x₀, x₁, subject to the tame relation τ^σ = τ², one explicit wild word relation, and the requirement that the closed normal subgroup generated by the wild generators x₀, x₁ be pro-2. A profinite group is a group constructed as a limit of finite groups, the natural algebraic structure for capturing symmetries of infinite field extensions.