Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
...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.
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...
