Exercise 8, § 5

Let $A$ be a ring, $P$ a monic polynomial of degree $n$ of $A[X]$, $B$ the $A$-algebra $A[X]/(P)$. We denote by $x$ the class of $X$ in $B$. Let $\ell : B \to A$ be the $A$-linear form such that $\ell(x^{n-1}) = 1$, $\ell(x^i) = 0$ for $0 \leq i \leq n-2$; we denote by $\tilde{\ell} : B[X] \to A[X]$ the $A[X]$-linear mapping deduced from $\ell$ by extension of scalars. a) Let $\sigma : B[X] \to A[X]$ be the $A[X]$-linear mapping such that $\sigma(x^i) = X^i$ for $i = 0, \ldots, n-1$. Prove the formula $P \tilde{\ell}(Q) = \sigma((X-x)Q)$ for every element $Q$ of $B[X]$ (verify it for $Q = 1, \ldots, x^{n-1}$). b) Let $P^*(X) = b_{n-1} X^{n-1} + \ldots + b_0$ be the polynomial of $B[X]$ such that we have $P(X) = (X-x)P^*(X)$ in $B[X]$. Prove that $(b_0 \ell, \ldots, b_{n-1} \ell)$ is the basis of $B^*$ dual to the basis $(1, \ldots, x^{n-1})$ of $B$ (apply $a$ to the polynomials $x^i P^*$). Deduce that the $B$-module $B^*$ is free of rank one, and that $\ell$ forms a basis of it. c) Prove the equality $\mathrm{Tr}_{B/A} = P'(x)\ell$ in $B^*$ (cf. exerc. 7 a)).