Exercise 2.4.10

Proof.

(a)

If $p$ is an irreducible factor in $F[{\sigma }_1,\cdots ,{\sigma }_n]$ which divides $C^m$ and $D^m$ ( $m\in {\mathbb{N}}^{\text{∗}}$), as $F[{\sigma }_1,\cdots ,{\sigma }_n]\simeq F[u_1,\cdots ,u_n]$ is a factorial domain, $p$ divides $C$ and $p$ divides $D$, which is in contradiction with the fact that $C,D$ are relatively prime in $F[{\sigma }_1,\cdots ,{\sigma }_n]$. Consequently $C^m,D^m$ are relatively prime in $F[{\sigma }_1,\cdots ,{\sigma }_n]$.

(b)

If $p$ is an irreducible factor in $F[x_1,\cdots ,x_n]$ which divides $C$ and $D$, then $C=\mathit{pE},E\in F[x_1,\cdots ,x_n]$. As $C$ is symmetric, we obtain, using 2.31: $$\begin{array}{lll}\hfill C=\sigma \cdot C=(\sigma \cdot p)(\sigma \cdot E).& & \hfill \text{(1)}\end{array}$$

Therefore $\sigma \cdot p$ divides $C$ for all $\sigma \in S_n$, and it is the same for $D$.

(c)

The product, for all $\sigma \in S_n$, of the relations (1) gives: $$C^{n!}=\underset{\sigma \in S_n}{\prod }\sigma \cdot p\underset{\sigma \in S_n}{\prod }\sigma \cdot E.$$

Therefore $P=\underset{\sigma \in S_n}{\prod <!--\; FUNCTION\; APPLICATION\; -->}\sigma \cdot p$ divides $C^{n!}$ in $F[x_1,\cdots ,x_n]$, and similarly for $D$.

By Exercise 2.2.7, $P$ is symmetric, and $C^{n!}=\mathit{PQ},D^{n!}=\mathit{PS},Q,S\in F[x_1,\cdots ,x_n]$.

As $C^{n!},D^{n!},P$ are symmetric, $Q,S$ are also symmetric. Indeed, for all $\sigma \in S_n$, $\mathit{PQ}=C^{n!}=\sigma \cdot C^{n!}=(\sigma \cdot P)(\sigma \cdot Q)=P(\sigma \cdot Q),$ thus $Q=\sigma \cdot Q$.

Therefore $P=P_1({\sigma }_1,\cdots ,{\sigma }_n)$, and $P_1\in F[{\sigma }_1,\cdots ,{\sigma }_n]$ divides $C^{n!},D^{n!}$ in $F[{\sigma }_1,\cdots ,{\sigma }_n]$. As the irreducible polynomial $p$ divides $P$, $P_1$ is not a constant. Therefore the two polynomials $C^{n!},D^{n!}$ are not relatively prime in $F[{\sigma }_1,\dots ,{\sigma }_n]$, and by (a), $C,D$ are not relatively prime in $F[{\sigma }_1,\dots ,{\sigma }_n]$, in contradiction with the hypothesis.

Conclusion : two relatively prime polynomials in $F[{\sigma }_1,\cdots ,{\sigma }_n]$ remain relatively prime when regarded as elements of $F[x_1,\cdots ,x_n]$.