Residue Classes

Question

I am trying to show that:

$$\sum\limits_{\beta \in \mathbb{Z}_p^*}{\beta^{-1}}=\sum\limits_{\beta \in \mathbb{Z}_p^*}{\beta}=0$$

Where p is an odd prime.

I really dont know where to start, but my best guess is that because B and the inverse of B should cancel out, then it should equal 0. Am I right in thinking that?

How would I go about proving this equation, I was thinking of using the additive inverse theorem.

Thanks

What's "the aditive inverse theorem"? – Gerry Myerson Feb 07 '13 at 05:09 — Gerry Myerson, Feb 07 '13 at 05:09

score 2 · Accepted Answer · answered Feb 07 '13 at 04:51

2

The following sets are identical:

$$\{x\in\Bbb Z_p^*\}=\{x^{-1}\;;\;x\in\Bbb Z_p^*\}$$

thus both sums in your link are identical. Finally

$$\sum_{x\in\Bbb Z_p^*}x=1+2+...+(p-1)=\frac{p(p-1)}2=0\pmod p$$

since $\,p\,$ is prime.

answered Feb 07 '13 at 04:51

DonAntonio

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"since $p$ is prime" --- indeed, since $p$ is odd. – Gerry Myerson Feb 07 '13 at 05:10

Residue Classes

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