Show that $\Bbb Z_p$ contains all the $(p-1)$th roots of unity. For which primes $p$ does $\Bbb Z_p$ contains primitive fourth roots of unity? Here $\Bbb Z_p$ is the ring of $p$-adic integers. Proving that it has a $(p-1)$th root of unity is easy, but all roots is another matter. Please help me with these questions.
I think for the second question, $p$ has to be $5$, but maybe there are other answers that I didn't think of.
Edit: I solved the first question, only need the second one. From what I can see, to be the primitive $4$th root, $4\mid(p-1)$, is that right?