Exercise 3, § 6

Let $f$ be a symmetric polynomial of $K[X_1, \ldots, X_n]$ and $\varphi$ the unique polynomial of $K[Y_1, \ldots, Y_n]$ such that $f(X_1, \ldots, X_n) = \varphi(s_1, \ldots, s_n)$. Show that the total degree of $\varphi$ is equal to the degree of $f$ in any one of the $X_i$.