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?

  • 2
    Another approach: since taking character groups is exact for (I think arbitrary, but certainly for) finitely generated Abelian groups, the left-hand side is $\bigl(\prod_{\chi \in \widehat{\langle a\rangle}} (1 - \chi(a)T)\bigr)^{[A \mathbin: \langle a\rangle]}$, which is easily seen to be as stated. – LSpice Oct 11 at 16:38
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    I think I first saw this in Lang's book on cyclotomic fields, perhaps in the early sections where he uses characters with Jacobi sums. Have you looked there or in Narkiewicz's massive tome on algebraic number theory? Anyway, I wonder if you really need to cite the literature since a proof is short. Since $a$ has order $h$, the mapping $\widehat{A} \rightarrow \mu_h$ by $\chi \mapsto \chi(a)$ is a surjective homomorphism, so each $h$th root of unity is a value $|\widehat{A}|/h = |A|/h = N/h$ times. Thus the product is $\prod_{z^h=1} (1-zT)^{N/h} = (1 - T^h)^{N/h}$, QED. – KConrad Oct 11 at 17:38
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    For $A=(\mathbb{Z}/m\mathbb{Z})^{\times}$, this appears explicitly in Serre's Course in Arithmetic (Chapter VI, Lemma 6). – so-called friend Don Oct 11 at 17:49
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    This only needs a word. It is merely a restatement of the fact, found in almost every representation theory text, that ${\rm Res}_{A}^{\langle a \rangle}(\rho_{A})=[A:\langle a\rangle]\rho_{\langle a \rangle}$, where $\rho_{B}$ denotes the regular representation of the group $B$ (applied in the case that $A$ is Abelian), – Geoff Robinson Oct 11 at 21:14
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    @LSpice yes, indeed it is. I had not read your argument closely before posting my reply. – KConrad Oct 11 at 21:30
  1. This is essentially proved in Rosen's "Number Theory in Function Fields", page 109, Lemma 8.14.

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  1. 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. enter image description here
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    If the aim is to have something to cite, then it seems that explaining why Lang's argument is actually about arbitrary Abelian groups (and specialises to the desired result) might take just as long as sketching out the very short proof. – LSpice Oct 11 at 17:25
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    Your citation to Rosen is not to a result directly about characters of general finite abelian groups (it is about a product of terms $1 - z^ku$ for $n$th roots of unity $z$ and varying $k$, with no characters visible), so I don't think it fits what the OP wants. – KConrad Oct 11 at 17:41
  • Thank you very much, but as @LSpice says I wanted something to cite. I think I will have to prove this in the paper. – EFinat-S Oct 12 at 15:28

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