On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$

Abstract

In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\ne u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4+4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2+6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after ruling out $2+6$, the patterns $2+2+4$, $2+2+2+2$, and $3+3+2$ regroup trivially to $2+6$ and are therefore impossible. Consequently, $P_{a,u}(t)$ admits no nontrivial factorization in $\mathbb{Z}[t]$.