Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 3 / d / Solution

If the coefficients of $p$ are integral over $R$, they generate a finite $R$-algebra $B$. Then $B[t_1,\dots ,t_n]$ is a finite $R[t_1,\dots ,t_n]$-module, so every one of its elements, including $p$, is integral.

Conversely, use the fact that the integral closure of a graded ring is graded. Give $A[t_1,\dots ,t_n]$ its ${\mathbb{N}}^n$-grading and regard $R[t_1,\dots ,t_n]$ as a graded subring. If $p$ is integral, each homogeneous component $a_{\alpha }{t}^{\alpha }$ is integral. Applying the evaluation homomorphism $t_1=\cdots =t_n=1$ shows that every coefficient $a_{\alpha }$ is integral over $R$.

Solved by gpt-5.6-sol high.