number theory about index

Asked Nov 14 '16 at 11:44

Active Nov 14 '16 at 11:50

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Given the congruence $x^3 \equiv a \pmod p$, where $p \geq 5$ is a prime and $\gcd(a,p)=1$, prove the following:

If $p \equiv1 \pmod 6$, then the congruence has either no solutions or three incongruent solutions modulo $p$
If $p \equiv5 \pmod 6$, then the congruence has a unique solution modulo $p$

Help me...

I don't know what I learn because Prof teaches so quickly.

So I need your help to organize my thoughts.

edited Nov 14 '16 at 11:50

barak manos

asked Nov 14 '16 at 11:44

文奭允

Read this for the case $a=1$ and theory. This is even more relevant. – Jyrki Lahtonen Nov 14 '16 at 11:56
And this recent thread is specific to exponent $3$, but the asker there made a bit more progress themselves than you have showed. – Jyrki Lahtonen Nov 14 '16 at 12:20

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