Exercise 7, § 10

Let N be a Galois extension of a field K such that $\mathrm{Gal}(N/K)$ is isomorphic to the symmetric group $S_n$ (V, p. 59, Example 5), n being a non-prime integer ; identify $\mathrm{Gal}(N/K)$ and $S_n$. Let $\Gamma_1$ be the subgroup of $S_n$ of order $(n-1)!$ leaving the number 1 invariant and let $\Gamma_2$ be the cyclic subgroup of $S_n$ of order n generated by the cycle $(1, 2, ..., n)$ (I, p. 142, Ex. 12). Let $E_1, E_2$ be the field of invariants of the subgroups $\Gamma_1, \Gamma_2$ respectively. Show that $E_1$ and $E_2$ are linearly disjoint over K, that there exists no field F other than K and $E_1$ such that $K \subset F \subset E_1$ but that there exist fields $F'$ distinct from $E_2$ and N such that $E_2 \subset F' \subset N$ (use Ex. 13 of I, p. 142 and Ex. 14, I, p. 143).