Exercise 5.4.1

Proof. $s(x)$ is defined in (5.18) by

Let $h=\underset{j=1}{\overset{m}{\prod <!--\; FUNCTION\; APPLICATION\; -->}}f(x-\lambda x_j)\in F[x_1,\dots ,x_m][x]$.

If $u=u(x_1,\dots ,x_m)\in F[x_1,\dots ,x_m]$, and $\sigma \in S_m$, we define $\sigma \cdot u=u(x_{\sigma (1)},\dots ,x_{\sigma (m)})$. If $v=u(x_1,\dots ,x_m,x)\in F[x_1,\dots ,x_m][x]$, where $v=\underset{i=0}{\overset{d}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{p}_i{x}^i,p_i\in F[x_1,\dots ,x_m]$, we write $\sigma \cdot v=\sum <!--\; FUNCTION\; APPLICATION\; -->(\sigma \cdot p_i)x^i$.

Then $\sigma \cdot (\tau \cdot v)=(\mathit{\sigma \tau })\cdot v$, and $\sigma \cdot (\mathit{vw})=(\sigma \cdot v)(\sigma \cdot w)$, for all $\sigma ,\tau \in S_n,v,w\in F[x_1,\dots ,x_m][x]$.

For every permutation $\sigma \in S_n$,

As $h=\underset{i=0}{\overset{d}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{p}_i(x_1,\dots ,x_m)x^i$ ( $d=\mathit{lm}$), and $g=\sigma \cdot h=\underset{i=0}{\overset{d}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}\sigma \cdot p_i(x_1,\cdots ,x_m)x^i$, every coefficient $p_i(x_1,\dots ,x_m)\in F[x_1,\dots ,x_m]$ is a symmetric polynomial.

The evaluation homomorphism $\phi$ defined by $x_1\mapsto {\gamma }_1,\dots ,x_m\mapsto {\gamma }_m$, where ${\gamma }_1,\dots ,{\gamma }_m$ are the roots of $h\in F[x]$ sends the coefficients of $h$ on the coefficients of $s$. Corollary 2.2.5 shows that $p_i({\gamma }_1,\dots ,{\gamma }_m)\in F,i=0,\dots d$, thus

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