Irreducibility of the Cuboid Polynomial $Q_{p,q}(t)$ via the Rank-One Chabauty-Coleman Method

Abstract

Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial, with coprime positive integers $p\ne q$. We prove that $Q_{p,q}(t)$ is irreducible over $\mathbb{Q}$, and hence over $\mathbb{Z}$. More generally, the normalized associated quintic $P_s(x)$ and its composition $P_s(t^2)$ are irreducible over $\mathbb{Q}$ for every $s\in\mathbb{Q}\setminus\{0,\pm1\}$. The proof reduces linear and quadratic factors to rational points and horizontal conjugate pairs on one fixed genus-$2$ curve. A certified rank upper bound for its Jacobian, together with an exact divisor calculation, gives a rank-one annihilating differential at the prime $3$. For the quadratic-factor case, the symmetric coordinates of a horizontal pair lie on an explicit double cover of an elliptic quartic. All rational affine points on this model are $3$-adically integral and reduce to six smooth residue classes. The pulled-back differential is a unit in each class, proving that six displayed points form the complete list. Their coordinates exclude every admissible parameter. A negative real root of the irreducible quintic then supplies the additional degree argument required for irreducibility of the degree-$10$ polynomial. The rank computation and all exact arithmetic certificates are specified explicitly for reproduction.