(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] cubic reciprocity?

I would like to prove the following conjecture:

If [tex]p \equiv 2\ (mod\ 3)[/tex] is a prime, then the cubing function [tex]x \mapsto x^3 [/tex] is a permutation of [tex]\mathbb{Z}_p [/tex].

I've tried to find a contradiction to the negation by assuming that if [tex]n \neq m\ (mod\ 3)[/tex], but [tex]n^3 \equiv m^3\ (mod\ 3)[/tex], then since [tex]m^3 - n^3 = (m-n)(m^2-mn+n^2)[/tex], we must have [tex](m^2-mn+n^2) \equiv 0\ (mod\ 3)[/tex] to avoid a contradiction concerning zero-divisors. Now I'm stuck.

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# Cubic reciprocity?

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