Explicit presentation of the absolute Galois group of Q₂
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What is an explicit marked profinite presentation of the absolute Galois group of Q₂? Generator and relation counts alone do not specify the tame and wild relations or identify the distinguished generators.
Why it matters and what counts as progress
Why it matters: A marked dyadic presentation fills the gap left by explicit presentations for odd primes and supports concrete local Galois calculations.
What counts as progress: Explicit generators, tame and wild-word relations, and the required pro-2 subgroup condition, with a proof that they present the intended Galois group.
How the problem developed
- August 2023 · Structural result [1]
Moshe Jarden and Mark Shusterman Computed the numbers of generators and relations needed for the absolute Galois group of any finite extension of Q_p, including Q₂, without giving the explicit marked presentation sought here.
- July 2026 · Explicit presentation [2]
David Roe and David Turturean Released an explicit four-generator presentation of G_Q₂ with two word relations and a pro-2 condition, together with a Lean proof of the presentation theorem.
AI contribution
AI-led work gives an explicit 2-adic absolute Galois presentation
Theorem 1.2 gives a marked profinite presentation of the absolute Galois group of Q₂ with four generators, a tame relation, an explicit wild-word relation, and a pro-2 condition on the normal subgroup generated by the wild generators.
Sources
- Moshe Jarden and Mark Shusterman, The absolute Galois group of a p-adic field
- David Roe and David Turturean, A presentation of the absolute Galois group of Q₂
- Dyadic Galois presentation project
- Lean formalizations of the dyadic presentation
- Roe formalization axiom census
- Development account for the dyadic presentation
- Finite-quotient verifier
Website timeline
Publication dates and editorial changes.
AI-led work gives an explicit 2-adic absolute Galois presentation