Primitive roots and indices Given the congruence $x^3 \equiv a$ (mod p), where $p \geq 5$ is a prime and gcd($a,p$)=1, prove the following  a. If $p \equiv 1$ (mod 6), then the congruence has either no solutions or three incongruent solutions modulo p.  b.If $p \equiv 5$ (mod 6), then the congruence has a unique solution modulo p.  How to solve it...by using the theory of indices.
number theory about index 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.
Doubt related to a theorem in congruences of Chapter 5 of Apostol Introduction to analytic number theory I am self studying number theory from Introduction to Analytic number theory by Apostol.   I have a doubt in one argument of proof of Theorem 5.14 and I am posting its image.      My doubt is in 2nd line of 2 nd paragraph how Apostol writes - If y is a solution of (5) then ay $\equiv $ at ( mod m) . How does y being a solution of (5) implies that?       Can someone please help