I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can understand the proof if I am willing to accept Stickelberger's theorem, but am having some trouble wrapping my head around the Theorem as a concept.

For example, in the proof, the author is using Stickelberger to prove Kronecker-Weber, yet in Washington's 'Introduction to Cyclotomic Fields', the formation of the theorem involves the fact that every abelian extension is contained in a cyclotomic field. (Kronecker-Weber)

The theorem states that the Stickelberger element, $$\theta=\sum_{a=1}^{p-1} a\sigma_a^{-1} \in \mathbb{Z}[\text{Gal}(F/\mathbb{Q})]$$ is an annhilator for the class group of $F$, where $p-1$ is the size of $\text{Gal}(F/\mathbb{Q})$.

I was wondering how we would apply this, for example, to $\mathbb{Q}(\sqrt{-33})$, where the galois group is clearly $\mathbb{Z}/(2)$, and we find the class group to be $\mathbb{Z}/(2)\:\text{x}\:\mathbb{Z}/(2)$, the Kleinian group. We have the ideal classes, $[1],[a],[b],[c]$ where $[a]$ can be represented by $(2,1+\sqrt{-33})$, $[b]$ can be represented by $(3,\sqrt{-33})$ and $[c]$ is represented by a non-principal ideal of norm $6$, but I can't quite figure out what this last ideal is explicitly.

In this example, can I apply the Stickelberger element to the class representatives, i.e. it is sufficient to show that $\theta(2,1+\sqrt{-33})$ is principal, and do the same for each ideal class?