Exercise 2, § 10

Let K be a field, $f_0$ an irreducible and separable polynomial in $K[X]$, of degree n ; let $\alpha_i$ ($1 \leq i \leq n$) be its roots in a splitting field N of $f_0$. Let F be the field of rational fractions $N(X_1, X_2, ..., X_n)$, E the subfield $K(X_1, X_2, ..., X_n)$ of F ; F is a Galois extension of E and $\mathrm{Gal}(E/F)$ is canonically isomorphic to $\mathrm{Gal}(N/K)$ (V, p. 71, Th. 5 ; V, p. 154, Ex. 8). Show that if $8 = \alpha_1 X_1 + \alpha_2 X_2 + ... + \alpha_n X_n$ then $F = E(8)$ and the minimal polynomial $g_1$ of 8 in $E[T]$ is an irreducible factor of the polynomial

Each permutation $\sigma \in S_n$, defines a K-automorphism of E, again denoted by a, by the condition $\sigma(X_i) = X_{\sigma(i)}$ for $1 \leq i \leq n$; show that $\mathrm{Gal}(N/K)$ is isomorphic to the subgroup of $S_n$, consisting of permutations a such that $\sigma(g_1) = g_1$. Further, if $f = g_1 g_2 ... g_r$ is a decomposition of f into irreducible factors in $E[T]$, show that for every index j such that $1 \leq j \leq r$ there exists a permutation $\sigma_j \in S_n$, such that $\sigma_j(g_1) = g_j$.