lecture-06-2019-03-29.tex 8.13 KB
 rrueger committed May 01, 2019 1 2 3 \setcounter{chapter}{5} \chapter{Solvability}  rrueger committed May 06, 2019 4 5 6 7 8 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.  rrueger committed May 01, 2019 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47  \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