Prove that $\Bbb F_p^\times$ is equal to Miller–Rabin primality test for prime number

Question

I want to prove, that $\Bbb F_p^\times = MRP(p)$.

I think, that I have to start with this statement: $\{a \in \Bbb F_p^\times | a^2 = 1 \} = \{1; -1\}$

But I do not know how to continue this idea.

Answer 1

A number $p>1$ is prime if and only if the only solutions of the equation $x^2\equiv 1\ (\ mod\ p\ )$ are {$-1,1$}

If only a weak pseudoprimetest is made (check, if $a^{p-1}\equiv 1\ (\ mod\ p\ )$) and $p$ passes this test, but the rabin-test fails for this base, then a nontrivial solution of $x^2\equiv 1\ (\ mod\ p\ )$ can be derived (showing that $p$ must be composite) What I do not know, if there is such a base $a$ for every composite number $p$.

If $p$ is prime, it passes the rabin-miller-test for every base $a$ coprime to $p$