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

The absolute Galois group of a field gathers all of its Galois theory into a single object: its closed subgroups correspond to the field's algebraic extensions, so a concrete description of the group is a concrete description of every extension at once. Number theorists use $p$-adic fields to collect congruence information into a convenient algebraic package, and they have played an increasingly important role in recent years, from the Langlands program to perfectoid spaces. For odd primes $p$, Jannsen and Wingberg gave explicit generators and relations for the absolute Galois group in the early 1980s, building on the work of Demushkin, Serre, and Labute on its maximal pro-$p$ quotient. The prime 2 resisted: Zel’venskiĭ described the Galois group of the maximal extension without simple ramification for 2-adic base fields of odd degree, and Diekert obtained structure results for dyadic fields under additional hypotheses. Jarden and Shusterman later proved that $G_{\mathbb{Q}_2}$ is finitely presented and determined the minimal numbers of generators and relations abstractly, but no explicit presentation of the absolute Galois group of $\mathbb{Q}_2$ (concrete generators and concrete relations) was known.

The missing dyadic case was recently posed as one of Epoch AI's FrontierMath open problems by the first author. This paper closes it, giving an explicit presentation in an enriched, marked sense that goes back to Jannsen and Wingberg's description for odd $p$. Beyond completing the local story, such presentations matter because these local absolute Galois groups fit into the more mysterious absolute Galois group of $\mathbb{Q}$, and an explicit presentation turns questions about extensions, their counts, and their symmetries into finite computations.

The absolute Galois group of $\mathbb{Q}_2$ is the profinite group generated by four marked elements $\sigma,\tau,x_0,x_1$, subject to the tame relation $\tau^\sigma=\tau^2$, one explicit wild word relation, and the requirement that the closed normal subgroup generated by the wild generators $x_0,x_1$ be pro-2. The precise statement is Theorem 1.2 of the paper.

The proof characterizes the finite quotients of the candidate group by admissible generating quadruples, then shows that the candidate and $G_{\mathbb{Q}_2}$ admit exactly the same number of surjections onto every finite group: the tame and maximal pro-2 quotients are matched in marked form, lifting problems through elementary abelian layers are compared (Fox–Jacobian calculus on the candidate side, local Tate duality on the Galois side, with a quadratic Gauss-sum comparison in the self-dual case), and an induction driven by Fourier inversion closes the count. Profinite reconstruction then gives the isomorphism.

The result has been checked at three levels. The manuscript carries the complete proof but has not yet been externally peer reviewed. Two separately initiated Lean 4 formalizations verify the theorem, Roe's down to nine named interfaces to the classical literature and Turturean's down to seven external inputs after a documented later adaptation. And the finite-quotient verifier confirms the predicted count of Galois extensions for all 5,402 finite test groups: strong evidence, though short of a proof by itself.