Exercise 26, § 1

a) Let $F \subset M(\Delta)$ be a differential field and let $K$ be a finite Galois extension of $F$, a subfield of a $M(\Delta_1)$. Let $t$ be a primitive or an exponential of a primitive of a function of $F$; show that if $t$ is transcendental over $F$ there is no function $u \in K$ such that $t' = u'$. (Consider separately the two cases where $t' = a \in F$ or $t' = bt$ with $b \in F$; obtain a contradiction by considering the transforms $\sigma.u$ of $u$ by the Galois group of $K$ over $F$ and their derivatives $\sigma.u'$; in the first case show that one would have $t' = c'$ for some $c \in F$, and in the second case, putting $N = [K : F]$, one would have $(t^N/c)' = 0$ for some $c \in F_*$)

b) Suppose that $t$ is transcendental over $F$ and that $t' = bt$ with $b \in F$; show that there does not exist any $c \neq 0$ in $K$ such that $(ct^m)' = 0$ (same method).*