inverse_galois_problem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

The inverse Galois problem: Is every finite group realizable as the Galois group of some Galois extension of ℚ? Proved for all solvable groups (Shafarevich), symmetric groups, many simple groups. The general case remains open.

Why this node was retired

The posted statement is

import Mathlib

theorem inverse_galois_problem (G : Type*) [Group G] [Fintype G] :
    ∃ (K : Type*) (hF : Field K),
      Nonempty (G →* Units (K →+* K)) ∧
      (∃ phi : G ≃ (K →+* K), Function.Injective phi) := by
  sorry

There is no extension of ℚ and the equivalence to endomorphisms is an equivalence of underlying sets, not a group isomorphism. The separate homomorphism need not be injective.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For every finite group G, conjecture a finite Galois extension K/ℚ and a group isomorphism G≃*Gal(K/ℚ). The extension must have characteristic zero and the isomorphism must respect group operations.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.