Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 10E / b / Solution

Regard the polynomial as an element of $\mathbb{C}[x][y]$:The prime $x-1$ divides every nonleading coefficient, its square does not divide the constant coefficient, and it does not divide the leading coefficient $x^3$. Eisenstein's criterion proves irreducibility over $\mathbb{C}(x)[y]$; the polynomial is primitive in $\mathbb{C}[x][y]$, so Gauss's lemma proves irreducibility there.