3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ In your example, the non-trivial Galois element acts trivially on the class group. Indeed, every non-trivial element of the class group has order 2, and so every element is its own inverse. But the Galois group sends an ideal to its conjugate, which is its inverse in the class group (since the product of an ideal and its conjugate is the norm of the ideal, which is principal). So in your example Stickelberger says that multiplication by 2 kills the class group. A better way of looking at this is that it is Stickelberger who tells you that the action of the Galois group on the class group is... $\endgroup$ – Alex B. Mar 1 '15 at 13:28
  • $\begingroup$ ...trivial. Indeed, the automorphism group of $C_2\times C_2$ is isomorphic to ${\rm GL}_2(\mathbb{F}_2)\cong S_3$, and if $\tau$ is an element of order 2 in the automorphism group, then $\text{id}+\tau$ does not kill $C_2\times C_2$. So it follows from Stickelberger that the non-trivial Galois element cannot correspond to an element of order 2 in the automorphism group of $C_2\times C_2$, therefore it must correspond to the identity automorphism. In general, Stickelberger tells you something about the structure of the class group as a Galois module. $\endgroup$ – Alex B. Mar 1 '15 at 13:31
  • $\begingroup$ @AlexB.Thanks for your response. I am a new user so I believe I can't upvote your comments yet but I will when I get enough rep. I understand your answer and see that it is a more general way of looking at this, but is it incorrect to consider that $\theta (2,1+\sqrt{-33}) = (2,1+\sqrt{-33}) + 2(2,1-\sqrt{-33})$ which is a principal ideal? I was just wondering if there would be a way to explicitly look at what the element does to a non-principal ideal. Also, what if the action of $G$ was non-trivial? $\endgroup$ – Ramified_Minds Mar 1 '15 at 14:07
  • $\begingroup$ Or perhaps I am thinking about it in too basic terms.. $\endgroup$ – Ramified_Minds Mar 1 '15 at 14:19

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.