\setcounter{chapter}{5}
\chapter{Solvability}
In this lecture we will investigate a very specific field extension, namely that
of the rational functions on $n$ variables $\C(X_1, \ldots, X_n)$ over the field
of rational functions in the elementary symmetric functions $\C(e_1, \ldots,
e_n)$. We have seen elementary symmetric functions before in some examples, but
now we will finally define them in general.
\section{Elementary Symmetric Functions}
\begin{definition}[Elementary Symmetric Functions]
Let $X_1, \ldots, X_n$ be variables. The set of \textbf{elementary symmetric
functions} $\set{e_0, \ldots, e_n }$ on these $n$ variables is defined as the
coefficients of the polynomial
%
\begin{align*}
\prod _{k=1} ^n (Z - X_k) = \sum _{k=0} ^n e_{n-k} Z^k
\end{align*}
%
we can also write them in a more concrete closed form:
%
\begin{gather*}
\begin{cases}
e_0 = 1 \\
e_1 = \sum _{i=1} ^n X_i \\
e_2 = \sum _{1 \leq i