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

This paper resolves the inverse Galois problem for the final sporadic finite simple group by proving that the Mathieu group $M_{23}$ occurs as a Galois group over $\mathbb{Q}$ , achieved through the construction of an explicit degree 23 polynomial using numerical Belyi map algorithms.

Original authors: Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, Shaowu Zhang

Published 2026-08-11

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CC BY 4.0

Original authors: Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, Shaowu Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master builder trying to construct a specific, incredibly complex castle. In the world of mathematics, this "castle" is a Galois group, which is essentially a set of rules describing how the pieces of a mathematical puzzle (specifically, the solutions to an equation) can be shuffled around without breaking the structure. For decades, mathematicians have been trying to build these castles using only the most basic materials: the rational numbers (fractions like 1/2 or 3/4). This challenge is known as the Inverse Galois Problem. It's like asking, "Can we build every possible type of castle using only these specific bricks?"

Most of the castles have been built. There are 26 special, rare types of these mathematical structures called "sporadic groups" that don't fit into any standard family. By the late 1980s, builders had successfully constructed 25 of them. But one stubborn, elusive castle remained unfinished: the Mathieu group M23. It was the last missing piece of the puzzle. The question wasn't just about building it; it was about proving it could exist using the standard materials of rational numbers. If it couldn't be built this way, the entire theory of how these structures relate to numbers would have a giant hole in it.

The Final Piece of the Puzzle

In this paper, a team of mathematicians finally hands over the blueprints for the last missing castle. They prove that the Mathieu group M23 can indeed be constructed as a Galois group over the rational numbers. To do this, they didn't just guess; they built an explicit, working model. They produced a specific polynomial (a giant mathematical equation) of degree 23 with rational coefficients. When you solve this equation, the collection of all its solutions forms a "splitting field," and the way these solutions can be shuffled around perfectly matches the rules of the M23 group.

The Journey: From "Almost" to "Exactly"

The path to this discovery was a bit like trying to find a needle in a haystack, but the haystack was made of mathematical symmetries. The team used a powerful tool called the rigidity method. Imagine you are trying to design a unique sculpture. If you have a set of constraints (like "it must have three specific bumps"), usually there are infinite ways to make it. But sometimes, the constraints are so tight that there is only one possible shape. In math, this is called "rigidity."

For the M23 group, the team tried to find a set of constraints that would force a unique shape. They looked at three specific types of symmetries (conjugacy classes) within the group. They hoped to find a "rigid triple" that would create a unique, one-of-a-kind mathematical cover (a map from one shape to another). However, when they ran the numbers, they found that this specific combination wasn't rigid enough to force a single solution. Instead, it opened up a door to seven different possible shapes.

Usually, this would be a dead end. If there are seven possibilities, how do you know which one is the "real" one defined over rational numbers? Here is where the story takes a miraculous turn. The team expected the symmetries of the rational numbers to shuffle these seven shapes around randomly. But, much to their surprise, they found that one of these seven shapes was a "fixed point." It stood still while the others moved. This specific shape was defined over a special number field involving the square root of -23, but it had a hidden property: it could be "descended" or pulled back down to the rational numbers.

The Construction: From Numbers to Polynomials

To turn this abstract shape into a concrete polynomial, the team used a high-tech digital tool called a Belyi map. Think of this as a sophisticated 3D scanner that takes a complex, curved surface and flattens it out into a map, revealing its hidden structure. They used numerical algorithms to calculate the coordinates of this shape with incredible precision.

Once they had the numbers, they didn't just trust the computer's floating-point decimals. They used a clever trick called the PSLQ algorithm to recognize the numbers as exact algebraic expressions. Miraculously, the coefficients of their curve turned out to live in the field $\mathbb{Q}(\sqrt{-23})$ . They then performed a series of algebraic maneuvers to construct a second function, $v$ , which acted as the key to unlocking the final polynomial.

The result was a degree 23 polynomial, $F(t, V)$ , which defines a regular extension of the rational numbers. By plugging in specific rational numbers for the variable $t$ , they generated specific polynomials with integer coefficients. One such polynomial, listed in the paper, looks like a chaotic mess of huge numbers:
$x^{23} - 184x^{21} - 1150x^{20} + \dots$
But when you solve this, the symmetry of the solutions is exactly the M23 group.

The Verdict

The authors are not just suggesting this might work; they have proven it. They used computer algebra systems (Magma and PARI/GP) to rigorously verify every step. They checked that the polynomial's splitting field has the correct Galois group and that it is unramified (smooth) outside a specific set of prime numbers {2, 3, 23}. They also confirmed that this construction works not just for one specific case, but implies that there are infinitely many such polynomials.

In short, the Mathieu group M23, the last of the 26 sporadic groups, has finally been realized over the rational numbers. The castle is built, the blueprint is complete, and the mathematical community now has the full set of 26 rare structures, all constructed from the same fundamental bricks.

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

Problem Statement
The Inverse Galois Problem asks whether every finite group occurs as the Galois group of a finite Galois extension of the rational numbers $\mathbb{Q}$ . While 25 of the 26 sporadic finite simple groups were realized as Galois groups over $\mathbb{Q}$ between 1984 and 1989, the Mathieu group $M_{23}$ remained the sole exception. Previous attempts to realize $M_{23}$ over $\mathbb{Q}$ failed, yielding realizations only over other fields (such as $\mathbb{Q}(\sqrt{-23})$ , $\mathbb{Q}(\sqrt{-7})$ , or specific number fields $F$ ) or failing to produce a regular extension of $\mathbb{Q}(t)$ . This paper resolves the problem by proving that $M_{23}$ is indeed a Galois group over $\mathbb{Q}$ .

Methodology
The authors employ a combination of the rigidity method, numerical Belyi map algorithms, and algebraic verification.

Failure of Standard Rigidity: The standard rigidity method relies on finding a triple of conjugacy classes $(C_1, C_2, C_3)$ in $M_{23}$ such that the associated Nielsen class has size 1 (a "rigid" triple). The authors demonstrate that for $M_{23}$ , no such rigid triple exists. The smallest Nielsen class size for any $G_\mathbb{Q}$ -stable multiset is 7, corresponding to the classes $\{2, 23A, 23B\}$ .
Non-Rigid Triple and Hurwitz Space: Instead of a rigid triple, the authors utilize the triple $\{2, 23A, 23B\}$ . The Riemann existence theorem implies the existence of exactly seven $M_{23}$ -covers of $\mathbb{P}^1_\mathbb{C}$ ramified over the set $S = \{0, \sqrt{-23}, -\sqrt{-23}\}$ with local monodromy in these classes. The quotient of such a cover by a point stabilizer yields a genus 4 curve $X_C$ .
Numerical Computation of Belyi Maps: Using the BelyiDB package (implementing algorithms by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by Hejhal and Stark), the authors numerically compute the seven curves $X_C$ associated with the Nielsen class. This involves computing power series expansions of differential forms to high precision.
Algebraic Recognition: The coefficients of the computed curves are recognized as algebraic numbers using the PSLQ algorithm. Remarkably, the authors find that the field of moduli for one of these seven curves is $\mathbb{Q}$ , despite the expectation that the Galois group $G_\mathbb{Q}$ would act transitively on the set of seven. This fixed point allows the curve to descend to a model $X$ defined over $\mathbb{Q}$ .
Construction of the Extension:
- The authors construct a canonical model of the genus 4 curve $X_K$ (where $K=\mathbb{Q}(\sqrt{-23})$ ) as a complete intersection of a quadric and a cubic in $\mathbb{P}^3$ .
- They compute rational functions $u$ and $v$ on $X$ to define a degree 23 map $t: X \to \mathbb{P}^1$ .
- By analyzing the function field $K(X)$ and applying linear algebra modulo small primes, they reconstruct a polynomial $F(T, V) \in \mathbb{Q}[T, V]$ of degree 23 in $V$ .
Verification of Galois Group:
- Geometric Monodromy: The local monodromy at the branch points (a 23-cycle at $\pm\sqrt{-23}$ and a specific cycle type at 0) implies that the geometric monodromy group $G_{geom}$ contains $M_{23}$ .
- Arithmetic Monodromy: By specializing the polynomial modulo 31 and using Magma to certify the arithmetic monodromy group is $M_{23}$ , they prove $G_{arith} = M_{23}$ . Since $M_{23}$ is not normal in any larger subgroup of $S_{23}$ , $G_{geom} = G_{arith} = M_{23}$ .

Key Contributions and Results

Theorem 1.1: There exists a Galois extension of $\mathbb{Q}$ with Galois group $M_{23}$ .
Theorem 1.3: There exists a regular Galois extension $L/\mathbb{Q}(t)$ with Galois group $M_{23}$ . This is a stronger result, implying the existence of infinitely many such extensions over $\mathbb{Q}$ via Hilbert's Irreducibility Theorem.
Explicit Polynomials: The authors provide explicit degree 23 polynomials with rational coefficients.
- Example 1.2: A polynomial $f(x)$ whose splitting field is an $M_{23}$ -extension of $\mathbb{Q}$ unramified outside $\{2, 3, 23\}$ .
- Example 3.7: A lower-height specialization $f(x)$ unramified outside $\{2, 7, 23\}$ .
Corollary 1.4: The results extend to any number field $k$ , proving the existence of regular $G$ -extensions of $k(t)$ and mutually independent $G$ -extensions of $k$ for any sporadic finite simple group $G$ .

Significance
The paper completes the program of realizing all 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ . The realization of $M_{23}$ was the final outstanding case. The work is significant not only for the result itself but for the methodology: it successfully navigates the failure of the standard rigidity method by leveraging a non-rigid triple where the Galois action on the Hurwitz space possesses a fixed point. This allows the descent of a cover defined over a number field to one defined over $\mathbb{Q}$ , a phenomenon the authors note they lack a conceptual explanation for, describing it as "miraculous."

The authors certify their claims using the Magma computer algebra system and provide their code publicly. They explicitly state that while AI tools were used for literature search, code generation, and proofreading, the final mathematical results and verifications were performed without AI assistance.

The Final Piece of the Puzzle

The Journey: From "Almost" to "Exactly"

The Construction: From Numbers to Polynomials

The Verdict

Technical Summary: The Mathieu Group M23M_{23}M23​ as a Galois Group over Q\mathbb{Q}Q

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