Example 3.7 — an explicit $M_{23}$ -polynomial unramified outside $\{2,7,23\}$

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MathieuM23.example_3_7

by Lucas · Oct 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

inverse-galois-problemmathieu-groupnumber-theory

Let $f=f_{3.7}\in\mathbb{Q}[x]$ be the degree-23 polynomial of Example 3.7 and let $K$ be a splitting field of $f$ over $\mathbb{Q}$ . Then $K/\mathbb{Q}$ is Galois, $\mathrm{Gal}(K/\mathbb{Q})\cong M_{23}$ , and $K$ is unramified outside $\{2,7,23\}$ :

\mathrm{Gal}(K/\mathbb{Q})\cong M_{23},\qquad p\mid d_K\ \Rightarrow\ p\in\{2,7,23\}.

Formalization Note As in Example 1.2: the statement covers every number field that is a splitting field of $f$ , and ramification is encoded through the absolute discriminant.

Preamble

import Definitions.Def_MathieuM23_Group
import Definitions.Def_MathieuM23_Polynomials

Formal statement

namespace MathieuM23

theorem example_3_7 (K : Type) [Field K] [NumberField K]
    [Polynomial.IsSplittingField ℚ K fEx37] :
    IsGalois ℚ K ∧ Nonempty ((K ≃ₐ[ℚ] K) ≃* M23) ∧
      ∀ p : ℕ, p.Prime → (p : ℤ) ∣ NumberField.discr K → p = 2 ∨ p = 7 ∨ p = 23 := by sorry

end MathieuM23

Source

X. Huang, B. Jackson, K.-H. Lee, B. Poonen, R. Pries, S. Zhang, *The Mathieu group M23 is a Galois group over Q*, arXiv:2608.08538v1 (2026), https://arxiv.org/abs/2608.08538, p. 7, Example 3.7

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Disclosure — NON-BLIND read-back. This read-back was written by the same agent that drafted the Lean statement (Aristotle, by Harmonic), with full knowledge of the source paper and of the intended meaning. It is not independent, blind testimony and must not be mistaken for an independent audit; a reviewer should compare it against the Lean code directly.

Statement. Let $K$ be any type in the lowest universe carrying a number field structure (characteristic $0$ , finite-dimensional over $\mathbb{Q}$ ). Assume $K$ is a splitting field over $\mathbb{Q}$ of the explicit polynomial $f_{3.7}=x^{23}+46x^{21}-598x^{20}+\dots-11217790920x+1243077066$ , with the full coefficient list in the polynomial definition file. Then:

$K/\mathbb{Q}$ is Galois;
there is an abstract group isomorphism $\mathrm{Aut}_{\mathbb{Q}}(K)\cong M_{23}=\langle g_1,g_2\rangle$ ;
every prime natural number $p$ dividing the absolute discriminant $d_K\in\mathbb{Z}$ equals $2$ , $7$ or $23$ .

Ramification is not mentioned directly. The hypotheses are satisfiable.

Human review

Endorsed by Shuze Chen · Oct 5, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 5, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Example 3.7 — an explicit M23M_{23}M23​-polynomial unramified outside {2,7,23}\{2,7,23\}{2,7,23}

Example 3.7 — an explicit $M_{23}$ -polynomial unramified outside $\{2,7,23\}$