# P-adic integers and roots of unity

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?

Hint: use Hensel's lemma. $\text{ }$

Second hint: what are the roots of $x^{p-1}-1$ in $\mathbf F_p$?
Third hint: for which $p$ does $x^2+1$ have a root in $\mathbf F_p$?