The Mathieu group $M_{23}$ is a Galois group over $\mathbb{Q}$

Abstract: Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984--1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we use a non-rigid triple of conjugacy classes of $M_{23}$ and compute Belyi maps to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark.

• The paper proves that M23 occurs regularly as a Galois group over Q(t), yielding infinitely many M23-extensions of Q by Hilbert irreducibility.
• The construction overcomes the absence of a rigid triple by using a seven-element Nielsen class whose arithmetic Galois action has a fixed point, enabling descent to Q.
• Certified monodromy computations and explicit degree-23 polynomials establish M23 extensions unramified outside {2,3,23} and {2,7,23}.

The Mathieu Group $M_{23}$ as a Galois Group over $\mathbb{Q}$

Main result and significance

The paper proves that the Mathieu group $M_{23}$ occurs as a Galois group over $\mathbb{Q}$, thereby completing the realization of all $26$ sporadic finite simple groups in the inverse Galois problem. More strongly, it constructs a regular $M_{23}$-extension of $\mathbb{Q}(t)$ and gives explicit degree-$23$ polynomials over $\mathbb{Q}$ whose splitting fields have Galois group $M_{23}$ (2608.08538).

The principal theorem is the existence of a regular Galois extension

$\mathbb{Q}$0

with Galois group $\mathbb{Q}$1. Hilbert irreducibility then yields infinitely many $\mathbb{Q}$2-extensions of $\mathbb{Q}$3. The paper also exhibits an explicit monic polynomial of degree $\mathbb{Q}$4 whose splitting field has Galois group $\mathbb{Q}$5 and is unramified outside $\mathbb{Q}$6. A second specialization, obtained after polynomial reduction, is unramified outside $\mathbb{Q}$7. These constructions provide concrete arithmetic realizations rather than merely abstract existence results.

The result resolves the last outstanding case in the long-standing program to realize every sporadic simple group over $\mathbb{Q}$8. Earlier constructions had realized $\mathbb{Q}$9 sporadic groups, while attempts for $M_{23}$0 produced extensions over quadratic or higher-degree number fields, or depended on geometric objects lacking rational points.

The obstruction to ordinary rigidity

The construction begins with the rigidity method for three-point covers. For a finite group $M_{23}$1 and prescribed conjugacy classes $M_{23}$2, the associated Nielsen class consists of generating triples

$M_{23}$3

satisfying $M_{23}$4, modulo simultaneous inner conjugation. A triple is rigid when the Nielsen class has cardinality one. In that case, the corresponding cover is defined over the field dictated by the Galois action on the conjugacy classes.

For $M_{23}$5, the authors calculate the Nielsen-class cardinalities for all Galois-stable multisets of three conjugacy classes. No such triple is rigid. Thus the standard rigidity method cannot directly produce a regular $M_{23}$6-extension over $M_{23}$7.

The smallest nonempty Nielsen class occurs for the multiset

$M_{23}$8

and has cardinality $M_{23}$9. The two classes of elements of order $\mathbb{Q}$0 are exchanged by the absolute Galois action through the quadratic field $\mathbb{Q}$1. Accordingly, the branch locus is taken to be the finite étale $\mathbb{Q}$2-scheme with geometric points

$\mathbb{Q}$3

and the local monodromy classes are assigned respectively to $\mathbb{Q}$4, $\mathbb{Q}$5, and $\mathbb{Q}$6.

This is the central structural feature of the argument: the construction does not rely on a rationally rigid triple. Instead, it exploits a non-rigid Nielsen class whose Galois action has a fixed point.

A fixed point in a seven-element Hurwitz scheme

The seven Nielsen classes correspond to seven $\mathbb{Q}$7-covers over $\mathbb{Q}$8 with the prescribed branch data. Quotienting each Galois cover by the point stabilizer $\mathbb{Q}$9 gives a degree-$26$0 cover

$26$1

The quotient curve $26$2 has genus %%%%38$M_{23}$39%%%%23$$-cycle monodromy there is one totally ramified point. The ramification contribution is therefore</p> <p>$$26$5

and Riemann–Hurwitz gives genus $26$6.

The decisive arithmetic phenomenon is that the absolute Galois group does not act transitively on the seven covers. It has a fixed point. The authors explicitly note that they do not have a conceptual explanation for this fixed point. It represents an $26$7-cover whose field of moduli is $26$8. Descent results for $26$9-covers then imply that the cover descends to $M_{23}$0.

This fixed-point mechanism replaces the uniqueness normally supplied by rigidity. It is also the paper’s most structurally striking claim: non-rigidity does not prevent descent when the arithmetic action on the relevant Hurwitz scheme has a rational point.

Numerical Belyi-map computation

To determine the cover explicitly, the authors use the numerical Belyi-map algorithm implemented in BelyiDB. The method computes expansions of holomorphic differentials on the genus-$M_{23}$1 curve from hyperbolic uniformization data. The relevant triangle group has signature $M_{23}$2, and a specified generating triple in $M_{23}$3 determines the monodromy representation.

The numerical computation yields expansions

$M_{23}$4

at a point above one of the branch points. These expansions show that the chosen point is not Weierstrass. The authors then test the canonical model and find that the curve is non-hyperelliptic. Its canonical embedding is consequently a complete intersection of a quadric and a cubic in $M_{23}$5.

Although the intermediate numerical calculations are not themselves treated as rigorous, the coefficients are recognized using PSLQ and subsequently verified algebraically. The resulting canonical equations unexpectedly have coefficients in

$M_{23}$6

rather than in a degree-$M_{23}$7 field suggested by the size of the Nielsen class. The paper describes this coefficient descent as another arithmetic coincidence requiring further conceptual explanation.

A rational function with divisor

$M_{23}$8

is first constructed on the curve over $M_{23}$9. After normalizing its branch values, it gives a Belyi-type map with branch values $\mathbb{Q}(t)$0, $\mathbb{Q}(t)$1, and $\mathbb{Q}(t)$2, which is then transformed to a map

$\mathbb{Q}(t)$3

branched at $\mathbb{Q}(t)$4, $\mathbb{Q}(t)$5, and $\mathbb{Q}(t)$6. To descend the degree-$\mathbb{Q}(t)$7 function field to $\mathbb{Q}(t)$8, the authors construct auxiliary functions over $\mathbb{Q}(t)$9 and combine them through the nontrivial automorphism of $23$0. This produces a polynomial

$23$1

of degree $23$2 in $23$3 and degree $23$4 in $23$5. The polynomial $23$6 defines the desired degree-$23$7 cover over $23$8.

The paper observes that degree $23$9 is not optimal: a degree-$\mathbb{Q}$0 function should exist over $\mathbb{Q}$1 by Riemann–Roch. However, degree $\mathbb{Q}$2 is the smallest possible degree for a nonconstant rational map defined over the relevant field. Degree-$\mathbb{Q}$3, $\mathbb{Q}$4, and $\mathbb{Q}$5 maps arise only from rulings on the defining quadric, and those rulings are not defined over $\mathbb{Q}$6 because the discriminant of the quadric is nonsquare.

Verification of the monodromy group

The geometric and arithmetic monodromy groups of $\mathbb{Q}$7 satisfy

$\mathbb{Q}$8

The local monodromy provides elements with cycle types:

• a $\mathbb{Q}$9-cycle above each of $M_{23}$0;
• cycle type $M_{23}$1 above $M_{23}$2.

The classification of transitive degree-$M_{23}$3 permutation groups implies that the geometric monodromy contains $M_{23}$4. To establish the reverse inclusion and identify the group exactly, the authors reduce modulo $M_{23}$5. The discriminant has the factorization

$M_{23}$6

where $M_{23}$7 and $M_{23}$8 is an irreducible polynomial of degree $M_{23}$9. The factor $\mathbb{Q}$00 reflects singularities of the chosen plane model rather than additional geometric ramification.

Because the relevant branch scheme remains étale modulo $\mathbb{Q}$01 and the extension is tame in characteristic $\mathbb{Q}$02, specialization of tame fundamental groups identifies the geometric monodromy in characteristic $\mathbb{Q}$03 with that in characteristic zero. A Magma computation gives

$\mathbb{Q}$04

The inclusions

$\mathbb{Q}$05

then force

$\mathbb{Q}$06

Since $\mathbb{Q}$07 is not normal in any larger subgroup of $\mathbb{Q}$08, the arithmetic monodromy is also $\mathbb{Q}$09. Equality of geometric and arithmetic monodromy proves regularity over $\mathbb{Q}$10.

This combination of geometric cycle data, classification of transitive permutation groups, tame reduction, and certified finite computation is especially effective here. Numerical reconstruction discovers the model, while the final group-theoretic and arithmetic assertions are verified exactly.

Explicit number fields

Specializing $\mathbb{Q}$11 to rational values produces infinitely many degree-$\mathbb{Q}$12 polynomials over $\mathbb{Q}$13 with splitting field Galois group $\mathbb{Q}$14. Polynomial reduction using PARI/GP yields smaller-height representatives.

One displayed polynomial has leading terms

$\mathbb{Q}$15

and splitting field unramified outside $\mathbb{Q}$16. Another explicit polynomial, substantially larger in height, is unramified outside $\mathbb{Q}$17. Both claims are certified computationally in Magma.

The ramification sets are arithmetically meaningful. The occurrence of $\mathbb{Q}$18 is forced by the $\mathbb{Q}$19-cycle inertia in the function-field construction, while the additional primes depend on the chosen specialization and the resulting discriminant. The examples therefore provide usable test cases for computations involving decomposition groups, resolvents, local ramification, and explicit representations of sporadic groups.

Consequences for the inverse Galois problem

The regular realization over $\mathbb{Q}$20 has consequences beyond the existence of one number field. For every number field $\mathbb{Q}$21, base change and specialization yield:

• a regular $\mathbb{Q}$22-extension of $\mathbb{Q}$23;
• an $\mathbb{Q}$24-extension of $\mathbb{Q}$25;
• an $\mathbb{Q}$26-extension of $\mathbb{Q}$27 linearly disjoint from a prescribed number field;
• infinitely many mutually independent $\mathbb{Q}$28-extensions of $\mathbb{Q}$29.

Together with earlier constructions for the other sporadic simple groups, this establishes the corresponding statements uniformly for every sporadic finite simple group.

The theoretical importance lies in extending the constructive inverse Galois toolkit beyond rational rigidity. The argument shows that a non-rigid Hurwitz space can still produce a rational regular cover when its Galois action possesses a fixed point. It also demonstrates that computational algebraic geometry can be integrated with descent, Nielsen-class analysis, and monodromy certification without treating numerical output as proof.

Computational methodology and future directions

The workflow combines several distinct computational layers: enumeration of Nielsen classes, numerical uniformization, Belyi-map reconstruction, PSLQ recognition, Riemann–Roch calculations, finite-field monodromy computation, and exact certification in Magma and PARI/GP. The authors explicitly distinguish heuristic numerical discovery from the algebraic verification that completes the proof.

This methodology suggests a broader role for automated systems in explicit arithmetic geometry. AI-assisted symbolic computation could help search Nielsen classes, identify promising branch data, optimize auxiliary functions, detect descent structures, and propose low-height specializations. However, the paper’s division between exploratory computation and certified proof remains essential. In particular, numerical recognition of algebraic coefficients and conjectural Galois actions must be followed by exact ideal, field, monodromy, and ramification computations.

Future work may seek a conceptual explanation for the fixed point in the seven-element Hurwitz scheme and for the unexpected descent of the canonical equations to $\mathbb{Q}$30. A more intrinsic description of the corresponding Hurwitz component could yield lower-degree or lower-height defining polynomials, including the degree-$\mathbb{Q}$31 model suggested by Riemann–Roch. The construction may also provide a template for non-rigid realizations of other groups whose conjugacy classes exhibit nontrivial cyclotomic Galois action.

Conclusion

The paper proves the existence of a regular $\mathbb{Q}$32-extension of $\mathbb{Q}$33 and supplies explicit degree-$\mathbb{Q}$34 polynomials over $\mathbb{Q}$35 with splitting field Galois group $\mathbb{Q}$36. Its method combines a non-rigid Nielsen class of size $\mathbb{Q}$37, a fixed point under arithmetic Galois action, numerical Belyi-map computation, algebraic descent, and certified monodromy calculations. The result completes the realization of the sporadic finite simple groups over $\mathbb{Q}$38 and establishes a technically significant model for explicit inverse Galois constructions beyond the rigid case.