Exercise 13.3.2

Proof. Let $G$ be the symmetry group of $\phi$ and ${\tau }_1,...,{\tau }_l$ be representatives for the left cosets of $G$ in $S_n$. The universal resolvent is $\Theta (y)={\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^l(y-({\tau }_i\cdot \phi )(x_1,...,x_n))$. Since $\phi \in \mathbb{Z}[x_1,...,x_n]$, for each $i,1\le i\le l$, $({\tau }_i\cdot \phi )(x_1,...,x_n)\in \mathbb{Z}[x_1,...,x_n]$, hence the coefficients of $\Theta (y)$ are in $\mathbb{Z}[x_1,...,x_n]$, i.e., $\Theta (y)\in \mathbb{Z}[x_1,...,x_n][y]$.

Suppose that $\sigma \in S_n$, then $(\sigma \cdot \Theta )(y)={\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^l(y-(\sigma {\tau }_i)\cdot \phi (x_1,...,x_n))$. But the set $\sigma {\tau }_1,...,\sigma {\tau }_l$ is also a set of left coset representatives of $G$ in $S_n$. Thus the application of $\sigma$ has merely permuted the roots of $\Theta (y)$ leaving the coefficients fixed. It means that coefficients of $\Theta (y)$ are symmetric and are polynomials in ${\sigma }_1,...,{\sigma }_n$ with integers coefficients (cf. Ex.9.1.6), i.e., $\Theta (y)\in \mathbb{Z}[{\sigma }_1,...,{\sigma }_n][y]$. The application of evaluation map $x_i\mapsto {\alpha }_i$ to $\Theta (y)$, so that ${\sigma }_i\mapsto c_i$, gives ${\Theta }_f(y)\in \mathbb{Z}[c_1,...,c_n][y]=\mathbb{Z}[y]$. □