Exercise 18, § 2

Notons $A$ (resp. $B$) the algebra $\mathbf{C} \{ z_1, z_2, z_3 \}$ (resp. $\mathbf{C} \{ u, v \}$) of convergent series in the variables $z_1, z_2, z_3$ (resp. $u, v$) and $\varphi : A \to B$ the unique algebra homomorphism such that $\varphi(z_1) = uv$, $\varphi(z_2) = v(e^u - 1)$, $\varphi(z_3) = v$. Now prove that $\varphi$ is injective, and that $\dim(A) = 3$ and $\dim(B) = 2$.*