Exercise 24, § 3

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

(b) Let $B$ be a factorial local integral domain of characteristic 2, $m$ its maximal ideal, $E$ its field of fractions and $A$ a derivation of $E$; suppose that the subfield $K$ of $E$, the kernel of $A$, satisfies $[E : K] = 2$; moreover, suppose that there exist two elements $x, y$ of $m$ such that $Ax$ and $Ay$ generate the ideal $q$ of $B$ generated by $\Delta(B)$. Show then that, if $t = \Delta z/z$ belongs to $q$, there exists an invertible element $u$ of $B$ such that $t = \Delta u/u$ (write $t = r\Delta x + s\Delta y$, where $r, s$ are in $B$, and use (a)).