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Galois' Professor's Revenge

M. Damele, A. Loi, M. Mereb, L. Vendramin

TL;DR

This work tackles the Inverse Galois Problem for Rubik-type groups by producing explicit polynomials whose splitting fields realize $\mathcal{R}_3$, $\mathcal{R}_4$, and $\mathcal{R}_5$ as Galois groups over $\mathbb{Q}$. It leverages the known structural decomposition of $\mathcal{R}_3$ and constructs higher-order realizations via fiber-product arguments, using $f$, $g$, and $h_i$-type polynomials subject to discriminant conditions to enforce the required group structure. The approach blends algebraic number theory with constructive polynomial methods and computational verification in Magma to confirm isomorphism with the target groups. The results provide explicit realizations of large puzzle-derived groups as Galois groups and offer a framework for extending these methods to even bigger groups via controlled disjoint extensions and sign-coordinated fiber products.

Abstract

We prove that the groups associated with the Revenge Cube and the Professor's Cube can be realized as Galois groups over the rationals.

Galois' Professor's Revenge

TL;DR

This work tackles the Inverse Galois Problem for Rubik-type groups by producing explicit polynomials whose splitting fields realize , , and as Galois groups over . It leverages the known structural decomposition of and constructs higher-order realizations via fiber-product arguments, using , , and -type polynomials subject to discriminant conditions to enforce the required group structure. The approach blends algebraic number theory with constructive polynomial methods and computational verification in Magma to confirm isomorphism with the target groups. The results provide explicit realizations of large puzzle-derived groups as Galois groups and offer a framework for extending these methods to even bigger groups via controlled disjoint extensions and sign-coordinated fiber products.

Abstract

We prove that the groups associated with the Revenge Cube and the Professor's Cube can be realized as Galois groups over the rationals.

Paper Structure

This paper contains 7 sections, 3 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The groups
  3. The Revenge Cube
  4. The Professor's Cube
  5. The proofs
  6. Magma code for verifying theorems
  7. Generators for the Professor's Cube group

Key Result

Theorem 1

Let where $f_8(X) = X^8-a X-b .$ Then the polynomial $fg$ has Galois group over the rationals isomorphic to the Rubik's Cube group $\mathcal{R}_3$ for a suitable specialization of parameters $a,b$ and $c.$

Figures (4)

  • Figure 1: The Rubik's Cube.
  • Figure 2: The Revenge Cube.
  • Figure 3: The Professor's Cube.
  • Figure 4: Unfolded net of Professor's cube, with labels. Our labeling is obtained by extending one particular labeling (slightly different from the one mentioned before) of the Revenge Cube, assigning numbers starting from 97 to the central cross of each face.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • proof : Proof of Theorem \ref{['thm:n=4']}
  • proof : Proof of Theorem \ref{['thm:n=5']}