Exercise 28, § 1

a) Let $F \subset M(\Delta)$ be a differential field. One says that a field $F' \supset F$ is an elementary extension of $F$ if there is a finite sequence

such that for every $j \leq n-1$ one has $F_{j+1} = F_j(t_j)$, where one has either $t_j' = a_j'/a_j$ for an $a_j \neq 0$ in $F_j$ (so that one can write $t_j = \log a_j$), or $t_j'/t_j = a_j'$ for an $a_j \in F_j$ (so that $t_j = \exp(a_j)$) An elementary function is a function that belongs to an elementary extension of $\mathbf{C}(z)$ (the field of rational functions over $\mathbf{C}$).

For example, the functions

are elementary functions.

b) Let $a \in F$ be such that there are elements $u_j$ ($1 \leq j \leq m$) and $v$ in an elementary extension $F'$ of $F$, and constants $\gamma_j \in \mathbf{C}$ ($1 \leq j \leq m$) such that

Show that then there are elements $f_j$ ($1 \leq j \leq p$) and $g$ in $F$ and constants $\beta_j \in \mathbf{C}$ ($1 \leq j \leq p$) such that

(Reduce, by induction on the length $n$ of the sequence (1), to the case where $F' = F(t)$. By modifying the $u_j$ first show that one can assume the $\gamma_j$ to be linearly independent over $\mathbf{Q}$. When $t$ is transcendental over $F$ use exerc. 27: if $t' = s'/s$ with $s \in F$ one must have $u_j \in F$ and $c \in F[t]$; by using () and arguing by contradiction show that one must have $v = \alpha t + b$ with $\alpha \in \mathbf{C}$ and $b \in F$. If $t'/t = r'$ with $r \in F$ show that by replacing $v$ by $v + k.r$ for a suitable integer $k \in \mathbf{Z}$, one can assume that $u_j \in F, \ v \in F[t]$; show by contradiction that $v$ is of degree 0, so $v \in F$. Finally, if $t$ is algebraic over $F$, embed $F'$ in a Galois extension $K$ of $F$ and consider the transforms of () by the Galois group of $K$ over $F$.)