The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:

Let $A$ be a finite abelian group of order $N$ and let $\hat A$ be its dual group. Let $a\in A$ have order $h$. Then $$\prod_{\chi\in\hat A}(1-\chi(a)T)=(1-T^h)^{N/h}.$$

I don't want to include a proof because one of the good things about this paper (I hope not the only one) is that is short.

I have searched in books about abelian groups, finite groups, representations, and number theory, but I could not find it. As usual, the only place I could find it is in one of the (magnificent) "blurbs" by Keith Conrad.

Does anyone knows a book where I can actually find this result?