Exercise 24, § 3

(a) Let $E$ be a field of characteristic 2, $\Delta$ a derivation of $E$, $K$ the subfield of $E$ kernel of $\Delta$; suppose that $[E : K] = 2$. Now prove that one has $\Delta^2 = a \Delta$ with $a \in K$, and that for an element $t \in E$ to be of the form $\Delta x/x$, it is necessary and sufficient that $\Delta t = at + t^2$.

(b) Let $B$ be a factorial local ring of characteristic 2, $m$ its maximal ideal, E its field of fractions, Δ a derivation of E; suppose that the subfield K of E, kernel of Δ, is such that [E : K] = 2; in addition, suppose that there exist two elements x, y of m such that Δx and Δy generate the ideal q of B generated by Δ(B). Now prove that if t = Δz/z belongs to q, there exists an invertible element u of B such that t = Δu/u (write t = rΔx + sΔy with r, s in B and use a)).