Exercise 25, § 1

The meromorphic functions on an open disc $\Delta$ in $\mathbf{C}$ form a field $M(\Delta)$. A subfield F of $M(\Delta)$ such that $u \in F$ implies $Du \in F$ is called a differential subfield of $M(\Delta)$.

a) For every polynomial $P(X) = X^m + a_1 X^{m-1} + \cdots + a_m$ with coefficients in $M(\Delta)$ show that there is an open disc $\Delta_1 \subset \Delta$ (not having the same centre as $\Delta$ in general) and a function $f \in M(\Delta_1)$ satisfying

at all the points $z \in \Delta_1$ where $f$ and the $a_j$ are holomorphic. If F is a subfield of $M(\Delta)$ containing the $a_j$ one says that the subfield of $M(\Delta_1)$ generated by $f$ and the restrictions to $\Delta_1$ of the functions of $F$ is the field $F(f)$ obtained by adjoining the root $f$ of $P$ to $F$. This abuse of language does not cause confusion because the restriction map $g \mapsto g|_{\Delta_1}$ of $F$ onto the subfield $M(\Delta_1)$ is injective. If $F$ is differential, so is $F(f)$.

b) For every function $a \in M(\Delta)$ show that there is an open disc $\Delta_2 \subset \Delta$ such that there is a function $g \in M(\Delta_2)$ satisfying the relation $g'(z) = a(z)$ at every point $z \in \Delta_2$ where $g$ and $a$ are holomorphic. If $F$ is a subfield of $M(\Delta)$ containing $a$ one says that the subfield $F(g)$ of $M(\Delta_2)$ generated by $g$ and the restrictions to $\Delta_2$ of the functions of $F$ is the field obtained by adjoining the primitive $\int a\,dz$ to $F$. If $F$ is differential, so is $F(g)$.

c) For every function $b \in M(\Delta)$ show that there is an open disc $\Delta_3 \subset \Delta$ such that there is an $h \in M(\Delta_3)$ satisfying the relation $h'(z) = b(z)h(z)$ at every point $z \in \Delta_3$ where $h$ and $b$ are holomorphic and $\neq 0$. If $F$ is a subfield of $M(\Delta)$ containing $b$, one says that the subfield $F(h)$ of $M(\Delta_3)$ generated by $h$ and the restrictions to $\Delta_3$ of the functions of $F$ is the field obtained by adjoining the exponential of the primitive $\exp (\int b\,dz)$ to $F$. If $F$ is differential, so is $F(h)$.

d) For every subfield $F$ of $M(\Delta)$ and every polynomial $P(X) = X^m + a_1 X^{m-1} + \cdots + a_m$ with coefficients in $F$ there is a field $K$ obtained by adjoining to $F$ successively the roots of polynomials such that $K$ is a Galois extension of $F$ and that one has $P(X) = (X - c_1) \ldots (X - c_m)$, where the $c_i$ are meromorphic functions (on a suitable disc) belonging to $K$. If $F$ is differential one has $(\sigma.g)' = \sigma.g'$ for every $g \in K$ and every element $\sigma$ of the Galois group of $K$ over $F_*$