Exercise 2, § 10

Let K be a commutative field of characteristic $\neq 2$, A and B two K-algebras and $\alpha$ a K-linear mapping of A into B; an $\alpha$-derivation of A into B is a K-linear mapping $d$ of A into B satisfying the relation

in other words, a derivation on the sense of Definition 1 with $A = A' = A''$, $B = B' = B''$, $d = d' = d''$, $\mu$ multiplication in A, $\lambda_1$ the mapping $(z, x) \mapsto z\alpha(x)$ of $B \times A$ into B and $\lambda_2$ the mapping $(x, z) \mapsto \alpha(x)z$ of $A \times B$ into B.

Show that if B is a commutative field, an extension of K, and $d \neq 0$, there exists $b \neq 0$ in B such that

If $b = c^2$ with $c \in B, c \neq 0$, there exist two homomorphisms $f, g$ of A into B such that $\alpha(x) = (f(x) + g(x))/2$, $dx = (f(x) - g(x))/2c$. Otherwise, there exists a quadratic algebra ($§ 2$, no. 3) B' over B, a $\mu \in B'$ such that $\mu^2 = c$, and a homomorphism $f$ of A into B' such that $\alpha(x) = (f(x) + \overline{f(x)})/2$, $dx = (f(x) - \overline{f(x)})/2$.