Exercise 18, § 8

Let K be a Henselian field with a valuation v, k the residue field of v and suppose that every finite algebraic extension of k is cyclic over k (which is for example the case when k is finite). Let A be the ring of v and f $\in A[X]$ a monic polynomial such that, iff $(X) = (X - \alpha_1) \ldots (X - a_n)$, where the $\alpha_i$ belong to an algebraic closure of K, the element $D = \prod_{i < j} (\alpha_i - \alpha_j)^2$ of A ("discriminant" off) is such that $v(D) = 0$. Let $\tilde{f}_j$ ($1 \leq j \leq s$) be the irreducible factors of the canonical image $\tilde{f}$ of f in $k[X]$ and let $r_j$ be the degree of $\tilde{f}_j$. Show that the Galois over K of the splitting field of f, considered as a group of permutations of the a, is generated by a permutation $\sigma$ which decomposes into cycles of respective lengths $r_1, r_2, \ldots, r_s$ (observe that there exists only a single extension of k of given degree, up to a k-isomorphism). Deduce that, for D to be a square in A, it is necessary and sufficient that $n - s$ be even ("Stickelberger's Theorem"; examine under what condition D is invariant under $\sigma$).