Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 3 / ii / Solution

Let $\mathfrak{m}$ be maximal in $\mathbb{Q}[T_1,\dots ,T_n]$ and putThe Zariski lemma makes $K$ a finite extension of $\mathbb{Q}$. Base change givesThis ring is nonzero because the field extension makes $\mathbb{C}$ a faithfully flat module over $\mathbb{Q}$. Choose a maximal ideal $\mathfrak{n}$ of the quotient, or equivalently a maximal ideal of $\mathbb{C}[T_1,\dots ,T_n]$ containing $\mathfrak{m}\mathbb{C}[T_1,\dots ,T_n]$. Its contraction to the rational polynomial ring contains $\mathfrak{m}$ and is proper, so maximality of $\mathfrak{m}$ forces