I am reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington and I need to prove certain identities for the orthogonal idempotents of a group ring (namely a,b,c,d in the attached image) which are left for the readers.
My approach to proving (a) $\varepsilon^2_\chi = \varepsilon_\chi$ is as follows: \begin{align} \varepsilon^2_\chi = \varepsilon_\chi \cdot \varepsilon_\chi = \frac{1}{|G|} \sum_{\sigma_1 \in G} \chi(\sigma_1){\sigma_1}^{-1} \cdot \frac{1}{|G|} \sum_{\sigma_2 \in G} \chi(\sigma_2){\sigma_2}^{-1} = &\frac{1}{|G|^{2}} \sum_{\sigma_1, \sigma_2 \in G} \chi(\sigma_1\sigma_2){\sigma_1}^{-1}{\sigma_2}^{-1} \end{align}
How do I solve further?
I am not very good at dealing with multiple summations and that is the reason I am having difficulties in evaluating this sum. Please suggest me what should I do.
A lot of times what you have to do with these is to reindex. Here, try letting $\sigma_3=\sigma_2\sigma_1$, and add over $\sigma_1$ and $\sigma_3$ instead. The sum becomes $$\frac{1}{|G|^2}\displaystyle\sum_{\sigma_1,\sigma_3\in G}\chi(\sigma_3)\sigma_3^{-1}=\frac{1}{|G|}\sum_{\sigma_3\in G}\chi(\sigma_3)\sigma_3^{-1}=\varepsilon_{\chi}$$because $\chi(\sigma_1\sigma_2)=\chi(\sigma_2\sigma_1)$ and $(\sigma_2\sigma_1)^{-1}=\sigma_1^{-1}\sigma_2^{-1}$.