Gauss Sums and Stickelberger’s Theorem

In the previous section we used (but did not prove) Stickelberger’s theorem, which provides a nontrivial annihilator for the class group. In this chapter we prove this theorem, in a stronger form: we define an ideal of the group ring ℤ [ G ] $$\mathbb{Z}[G]$$ (where G is the Galois group), called Stickelberger’s ideal, and show that all its elements annihilate the class group. The proof relies on properties of Gauss sums, an arithmetical object interesting by itself. We develop the theory of Gauss sums to the extent needed for the proof of Stickelberger’s theorem. In the final sections we provide deeper insight into the structure of Stickelberger’s ideal. We determine its ℤ $$\mathbb{Z}$$ -rank, find a free ℤ $$\mathbb{Z}$$ -basis, study its real and relative parts, and prove Iwasawa’s class number formula.