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Let $w$ be a primitive $p^{th}$ root of unity, where $p$ is prime . Let $I,J \subset F_{p}$ where $F_{p}$ is a field of $p$ elements. Chebotarev's theorem states that if $I$ and $J$ have equal cardinality then the matrix $(w^{ij})_{i \in I, j\in J}$ has non-zero determinant.

What is the intuition behind this result? Note that it does not hold in general if $p$ is not prime. For example if $p=4$ then $I=\left\{0,2\right\}$ and $J=\left\{0,2\right\}$ gives a counterexample. Intuitively why doesn't it hold in general when $w$ is a primitive $n^{th}$ root of unity, $n$ being composite?

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    I only knew [Chebotarev's density theorem](http://en.wikipedia.org/wiki/Chebotarev's_density_theorem) but now I see there's a [Chebotarev's theorem on roots of unity](http://arxiv.org/abs/math/0312398). Thanks for educating me.2011-05-27
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    Why do you ask why it *doesn't* hold? Do you have an actual counterexample? From a very quick google search, the paper http://docs.google.com/viewer?a=v&q=cache:2atQ0-4p-LcJ:emis.impa.br/EMIS/journals/INTEGERS/papers/h18/h18.pdf+chebotarev+lenstra+stevenhagen&hl=en&gl=us&pid=bl&srcid=ADGEESjPPlf4jrFrGO_ZxI9neWUtBzMbsypRdORqYee0lcmkOrwi9WXewKkvOecXmGUc9X7URK2mwH3X2YBKwLghwjf3yUzEUkQZJL-e9OCph3Qsrs1ULXozhT_ZQsckAYrOr5xX4CTZ&sig=AHIEtbQMd3CC9xkLJH8QCYvTxR7aiAaefA gives an analogue for composite n.2011-05-27
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    See the 4th page of the paper on Chebotarev by Lenstra and Stevenhagen for a description of Chebotarev's proof for roots of unity with prime order p, which involves p-adic completions. They give some intuition.2011-05-27
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    @lhf: I was also going to comment the same :). Good to see our minds thinking along the same lines.2011-05-27
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    A counterexample for the compositie case would be if $w$ is a primitive 4-th roots of unity. Then $I=\left\{0,2\right\}$ ,$J=\left\{0,2\right\}$ gives a counter example.2011-05-27
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    By the way, the proof in the second link in the comment by lhf shows an important aspect of n-th roots of unity that distinguish the case of n being a prime power from other n: if w is a root of unity of order n then 1-w is a unit in Z[w] when n has more than one prime factor but 1-w is not a unit when n is a prime power (just one prime factor).2011-05-27
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    Myke: please put the counterexample into the question itself so readers will know there is something different in the case of prime vs. non-prime n.2011-05-27

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