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Elementary characterization of the Galois groups of $x^{12}+ax^6+b$

This paper provides an elementary characterization of all sixteen possible Galois groups of the irreducible polynomial $x^{12}+ax^6+b$ over $\mathbb{Q}$ , demonstrating that the group is uniquely determined by the Galois groups of the associated polynomials $x^4+ax^2+b$ and $x^6+ax^3+b$ along with checking whether at most two specific expressions in $a$ and $b$ are rational squares.

Original authors: Malcolm Hoong Wai Chen

Published 2026-08-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Malcolm Hoong Wai Chen

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 detective trying to solve a mystery, but instead of looking for fingerprints or footprints, you are looking for the hidden "personality" of a mathematical equation. This field is called Galois theory, and it lives in the world of algebra, where numbers and symbols dance together to form polynomials. Think of a polynomial like $x^2 + 5x + 6$ as a locked box. Inside the box are the "roots," which are the special numbers that make the equation equal zero. Sometimes, these roots are simple and easy to find, like $-2$ and $-3$. Other times, they are wild, messy, and tangled up in a way that makes them impossible to pull apart without breaking the rules of the number system we are using.

The "Galois group" is the name we give to the specific set of rules that describe how these roots can be shuffled around without breaking the equation. It's like a secret code that tells you how symmetrical the roots are. If you know the code, you know everything about the equation's structure. Mathematicians have been cracking these codes for centuries, but the equations get harder and harder as they get bigger. A simple equation might have a few roots, but a "degree 12" equation has twelve roots, creating a massive, complex web of possibilities. Understanding these groups helps mathematicians build better encryption for computers, design secure communication systems, and simply satisfy a deep human curiosity about the fundamental patterns of numbers.

In this paper, Malcolm Hoong Wai Chen tackles a very specific, tricky locked box: the equation $x^{12} + ax^6 + b$ . This isn't just any equation; it's a "power compositional" polynomial, which means it's built by plugging one shape into another, like a Russian nesting doll. Specifically, this equation can be seen as a 12th-degree monster, but it's also secretly a 4th-degree equation cubed, or a 6th-degree equation squared. Because it has these two different "identities," it offers a unique shortcut to solving the mystery.

The author's main finding is a complete, step-by-step guide to identifying the exact Galois group of this equation. Before this paper, knowing the group for such a complex equation was like trying to guess a person's entire personality just by looking at their shadow. Chen shows that you don't need to guess. Instead, you only need to look at two smaller, simpler versions of the equation (the 4th-degree and 6th-degree "shadows") and check if a few specific numbers are "perfect squares." If you know the groups of the smaller equations and run these simple square tests, you can pinpoint exactly which of the sixteen possible Galois groups the big equation belongs to. It's like having a master key that opens any of the sixteen different locks this equation could have.

The paper is very sure of its results; it doesn't just suggest or guess. It provides a rigorous mathematical proof that covers every single possibility. The author explicitly rules out certain combinations of the smaller groups, showing that some pairs of "shadows" simply cannot exist together for this type of equation. For example, the paper proves that you will never find a specific mix of a 4th-degree group and a 6th-degree group that leads to a contradiction. By eliminating the impossible, the author narrows down the field until only the correct answer remains for every case.

To make this work, the author uses a clever strategy. First, they list all the possible Galois groups that could fit the equation based on the smaller pieces. Then, they use a tool called a "linear resolvent," which is like a special scanner that looks at how the roots interact with each other. By checking if certain expressions involving the numbers $a$ and $b$ are rational squares (meaning they are the result of multiplying a fraction by itself), the author can distinguish between the remaining possibilities. The paper even provides a "reference sheet" (an algorithm) that anyone can follow. You plug in your numbers, check a few boxes, and the algorithm tells you the answer, whether it's group "12T11" or "12T81."

The paper also goes a step further by providing real-world examples. It doesn't just say "this is possible"; it writes out actual equations with specific numbers for $a$ and $b$ that create each of the sixteen different Galois groups. This proves that every single one of these sixteen outcomes is not just a theoretical possibility, but something that actually happens in the world of numbers. It's a complete map of the territory, showing that no matter what numbers you choose for $a$ and $b$ , you will always land in one of these sixteen specific, well-defined territories, and you will always know exactly which one it is.

Technical Summary: Elementary Characterization of the Galois Groups of $x^{12} + ax^6 + b$

Problem Statement
The paper addresses the problem of characterizing the Galois group of an irreducible polynomial $f(x) = x^{12} + ax^6 + b \in \mathbb{Q}[x]$ using its coefficients $a$ and $b$ . While complete characterizations exist for lower-degree power compositional polynomials (e.g., $x^4 + ax^2 + b$ and $x^6 + ax^3 + b$ ) and specific higher-degree cases, a streamlined, complete classification for the dodecic (degree 12) case $x^{12} + ax^6 + b$ had not previously appeared in the literature in this specific form. The polynomial is notable because it can be viewed as a power compositional polynomial in two distinct ways: $f(x) = g_4(x^3)$ where $g_4(x) = x^4 + ax^2 + b$ , and $f(x) = g_6(x^2)$ where $g_6(x) = x^6 + ax^3 + b$ .

Methodology
The author employs a combination of classical Galois theory, linear resolvents, and the analysis of subfield contents to derive the classification. The methodology proceeds as follows:

Decomposition and Subfield Analysis: The Galois group $G_{12}$ of $f(x)$ is analyzed in relation to the Galois groups $G_4$ and $G_6$ of the constituent polynomials $g_4(x)$ and $g_6(x)$ . The paper establishes that $G_{12}$ is constrained by the subfield content of $f(x)$ , which must include the Galois groups of the normal closures of intermediate fields corresponding to $g_4$ and $g_6$ .
Order Bounds and Discriminant Criteria: Using the structure of the roots (involving cube roots of roots of $g_4$ and square roots of roots of $g_6$ ), the author derives order bounds for $G_{12}$ , specifically $|G_{12}| \leq \min\{18|G_4|, 4|G_6|\}$ . Additionally, the discriminant of $f(x)$ is linked to the rationality of $b$ , determining whether $G_{12}$ is a subgroup of the alternating group $A_{12}$ .
Linear Resolvents: To distinguish between Galois groups that share the same order and subfield content (specifically the pairs $(12T12, 12T13)$ and cases involving $6T3$ and $6T9$ ), the paper utilizes linear resolvents corresponding to the functions $x_1 + x_2$ and $x_1x_2$ . The degrees of the irreducible factors of these resolvents serve as invariants to differentiate the transitive subgroups of $S_{12}$ .
Square Testing in Extension Fields: The classification relies on testing whether specific expressions involving $a$ and $b$ are squares in $\mathbb{Q}$ or in the field $\mathbb{Q}(\theta)$ , where $\theta$ is a root of $f(x)$ . Proposition 3.3 reduces the condition $r \in \mathbb{Q}(\theta)^2$ to checking if $r(a^2-4b)$ , $r(-a+2\sqrt{b})$ , or $r(-a-2\sqrt{b})$ are rational squares.

Key Contributions and Results
The paper provides a complete, elementary characterization of all sixteen possible Galois groups for $f(x)$ . The main result, Theorem 3.8, presents an algorithm that uniquely determines the Galois group $G_{12}$ based on:

The Galois groups $G_4$ and $G_6$ of the associated polynomials $g_4$ and $g_6$ .
The rationality of specific expressions involving $a$ and $b$ (e.g., $b$ , $b(a^2-4b)$ , $3(4b-a^2)$ , $-3b$ , and $3b(4b-a^2)$ ).
The reducibility of the auxiliary cubic polynomial $r(x) = x^3 - 3bx + ab$ .
The rationality of square roots of expressions like $3(a \pm 2\sqrt{b})$ .

The paper demonstrates that the pair $(G_4, G_6)$ uniquely identifies $G_{12}$ in most cases. However, for specific pairs where $G_4 \in \{4T2, 4T3\}$ and $G_6 \in \{6T3, 6T9\}$ , further testing of whether at most two specific expressions are rational squares is required to distinguish between multiple candidates (e.g., distinguishing $12T3$ from $12T10$ , or $12T12$ from $12T13$ ).

Numerical examples are provided in Table 2 to verify that each of the sixteen identified Galois groups actually occurs for some choice of rational coefficients $a$ and $b$ .

Significance and Scope
The author claims that this work contributes to a broader program aimed at systematically studying the Galois groups of polynomials of the form $x^{2m} + ax^m + b$ for composite $m$ . The classification is presented as a "stepping stone" for such studies, complementing existing work on prime $m$ .

The paper highlights the practical value of this classification by noting its application in recent work [11], which utilized these results to completely classify monogenic polynomials of the form $x^{12} + ax^6 + b$ . The author emphasizes that the classification is "streamlined" and "elementary," relying on explicit criteria rather than complex abstract machinery, making the determination of the Galois group accessible through direct computation of rational squares and polynomial reducibility.

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