Exercise 43, § 1

Let $E : U(A) \to \Lambda(A)$ be the mapping defined by

for $a \in U(A)$. a) Prove the commutativity of the diagram

Deduce that if $A$ is a $\mathbf{Q}$-algebra, one has

for every $a \in U(A)$. b) Now prove that $E$ is bijective. (If one puts $\prod_{n \in J} (1 - a_n(-T)^n) = \sum_{n \geq 0} c_n(a)(-T)^n$, it is enough to apply Exercise 33, a), p. 51 to the family of polynomials $c_n(X)$ in $\mathbf{Z}[X]$.) c) Endow $\Lambda(A)$ with the unique ring structure such that $E$ is a ring isomorphism. Now prove that the addition of $\Lambda(A)$ is the multiplication of series, with identity element 1. The multiplication of the ring $\Lambda(A)$, denoted $(f, g) \mapsto f * g$, is defined by the formula for all $a, b \in U(A)$; its identity element is $1 + T$. d) Let $\tau : A \to U(A)$ be the mapping defined in Exercise 37, c), p. 53. One has for every $a, b$ in $A$ and $f(T)$ in $\Lambda(A)$. e) Let $f$ and $g$ be two elements of $\Lambda(A)$ which are polynomials in $T$, and let $m$ be the degree of $f$. Put and These are polynomials in $X$ with coefficients in $A[T]$; $\varphi$ is monic and $f * g$ is the resultant $\operatorname{res}(\varphi, \gamma)$ of the polynomials $\varphi$ and $\gamma$ (cf. A, IV, p. 75). (One may start from the formula deduce the result if $A = \mathbf{Z}[X_1, ..., X_m]$ and $f = \prod_{i=1}^m (1 + X_i T)$, and pass from there to the general case.)