Exercise 27, § 1

Let $F \subset M(\Delta)$ be a differential field containing $\mathbf{C}$, and $t$ a primitive or an exponential of a primitive of a function in $F$, and suppose that $t$ is transcendental over $F$. Further, let $c_1, \ldots, c_n$ be elements of $F$ which are linearly independent over the field of rational numbers $\mathbf{Q}$; so if $u_1, \ldots, u_n$ and $v$ are elements of the field $F(t)$ and if

belongs to the ring $F[t]$, then necessarily $v \in F[t]$. Further, if $t \in F$, one necessarily has $u_j \in F$ for $1 \leq j \leq n$; if $t'/t \in F$ there exists for each $j$ an integer $v_j \geq 0$ such that $u_j/t^{v_j} \in F$. (Decompose the $u'_j/u_j$ and $v$ into simple elements in a suitable Galois extension of $F$ and use exerc. 26).*