Exercise 2.2.8

Question

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In this exercise, you will prove that if $\varphi \in F(x_1,\ldots,x_n)$ is symmetric, then $\varphi$ is a rational function in $\sigma_1,\ldots,\sigma_n$ with coefficients in $F$. To begin the proof, we know that $\varphi = A/B$, where $A$ and $B$ are in $F[x_1,\ldots,x_n]$. Note that $A$ and $B$ need not be symmetric, only their quotient $\varphi = A/B$ is. Let
$$C = \prod_{\sigma \in S_n \setminus \{e\} } \sigma \cdot B,$$
where we are using the notation of Exercise 7.
\begin{enumerate}
\item[(a)] Use Exercise 7 to show that $BC$ is a symmetric polynomial.
\item[(b)] Then use the symmetry of $\varphi = A/B$ to show that $AC$ is a symmetric polynomial.
\item[(c)] Use $\varphi = (AC)/(BC)$ and theorem 2.2.2 to conclude that $\varphi$ is a rational function in the elementary symmetric polynomials with coefficients in $F$.
\end{enumerate}

Richard Ganaye · Accepted Answer

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\begin{proof}
Let $\varphi = A/B\in F(x_1,\cdots,x_n)$ a symmetric rational function:
$$\forall \sigma \in S_n,\  \sigma \cdot \varphi = \sigma \cdot A/\sigma \cdot B = \varphi = A/B.$$
\begin{enumerate}
\item[(a)]
Let $$C =  \prod_{\sigma \in S_n\setminus \{e\}} \sigma \cdot B.$$
Then $$ BC = \prod_{\sigma \in S_n}\sigma \cdot B.$$
By Exercise 2.2.7, $BC$ is then a symmetric polynomial.

\item[(b)]
Note that the rules (2.31) for polynomials extend to rational functions. In particular, if $\varphi = A/B,\psi =A_1/B_1 \in F(x_1,\cdots,x_n)$, and $\sigma \in S_n$, 
$$\sigma \cdot (\varphi \psi) = (\sigma \cdot \varphi)\  (\sigma \cdot \psi).$$
Indeed,

$$ (\sigma \cdot \varphi)\  (\sigma \cdot \psi) = \frac{\sigma \cdot A}{\sigma \cdot B} \frac{\sigma \cdot A_1}{\sigma \cdot B_1}=\frac{\sigma \cdot (AA_1)}{\sigma \cdot (BB_1)} = \sigma \cdot (\varphi \psi). $$
Using this property, for all $\sigma \in S_n$, from $AC = \varphi BC$, we obtain

$$\sigma \cdot (AC) = (\sigma \cdot  \varphi)(\sigma \cdot (BC)) = \varphi BC = AC.$$
So  $AC$ is a symmetric polynomial.

\item[(c)]
So $\varphi = \frac{AC}{BC}$ is  the quotient of two symmetric polynomials, thus there exists $h,k \in F[x_1,\cdots,x_n]$ such that
$$\varphi = \frac{AC}{BC} = \frac{h(\sigma_1,\cdots,\sigma_n)}{k(\sigma_1,\cdots,\sigma_n)} =\left( \frac{h}{k}ight)(\sigma_1,\cdots,\sigma_n).$$
$\varphi \in F(\sigma_1,\cdots,\sigma_n)$ is a rational function in the elementary symmetric polynomials with coefficients in $F$.
\end{enumerate}
\end{proof}

Exercise 2.2.8

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Exercise 2.2.8

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