Exercise 4, § 2

Let $A$ be an integrally closed local ring, $\mathfrak{m}_A$ its maximal ideal and $K$ its field of fractions. For every element $x$ of $K$, denote by $A[x]$ the $A$-subalgebra of $K$ generated by $x$. Let $x$ be an element of $K^*$ such that $x^{-1}$ does not belong to $A$. Let $A[X]$ be the ring of polynomials in one indeterminate $X$, with coefficients in $A$, and let $\varphi$ be the evaluation mapping $P \mapsto P(x)$ of $A[X]$ onto $A[x]$. a) The kernel of $\varphi$ is the ideal $Q$ generated by the polynomials of the form $aX + b$, where $a, b \in A$ and $ax + b = 0$. (If $P(X) = \sum_{i=0}^{n} a_i X^i$ is such that $P(x) = 0$, note that then $-b_0 = a_0 x$ is integral over $A$ and hence belongs to $A$.) b) The ideal $\mathfrak{m}_A A[x]$ of $A[x]$ is prime, and it is maximal if and only if $x$ belongs to $A$.