Exercise 22, § 11

a) Let $k$ be a field, $f$ a separable monic polynomial in $k[X]$, $D$ a splitting extension of $f$, $x_1, \ldots, x_n$ the roots off in $D$ and $G$ the Galois group of $D$ over $k$. Assume that $G$ is cyclic. Show that for every element of $G$ to induce an even permutation of the $x_i$ it is necessary and sufficient that the number of irreducible factors of $f$ should be congruent to the degree of $f \mod 2$ (consider a generator $\sigma$ of $G$ and the decomposition into cycles of the permutation of the $x_i$ induced by $u$). b) Suppose further that $k$ is of characteristic $\neq 2$. Show that the two above conditions are equivalent to the fact that the discriminant of $f$ should be a square in $k$ (« Stickelberger's theorem »).