TITLE: Reference request: an elementary result on characters of finite abelian groups
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: This is essentially proved in Rosen's "Number Theory in Function Fields", page 109, Lemma 8.14.



This is also proved in Lang's "Algebraic Number Theory" (2nd edition), page 230. It is the equation with (*) in its beginning. The context is abelian extensions, so some of the notation make it seem number-theoretic, but the argument is general.