Exercise 1, § 10

a) Let K be a field, f a separable polynomial in $K[X]$ and N a splitting field of f. Show that for the group Gal(N/K) to operate transitively on the set of roots of f in N it is necessary and sufficient for f to be irreducible in $K[X]$. b) Suppose further that f is irreducible in $K[X]$; let x be a root of f in N. For there to exist no field E such that $K \subset E \subset K(x)$, distinct from K and $K(x)$ it is necessary and sufficient that Gal(N/K), qua transitive permutation group of the set of roots of f in N, should be primitive (I, p. 142, Ex. 13).

EXERCISES