I have two questions in algebraic number theory which I have difficulty understanding. I would be grateful if something could help me.
Let $K$ be an algebraic number field and let $O_K$ be its ring of integers. In addition, let $P$ be an (integral) prime ideal (in $O_K$) lying above the rational number $p$ and let $e>1$ be its ramification index. Now,
(1) Is there a connection between the fact that $K$ contains a primitive $p^{\rm th}$ root of unity (call it $\zeta_p$) and the divisibility of $e$ by $p-1$? I guess that if $\zeta_p\in K$ then $p-1\mid e$. What can we say about the ramification index in the case where $\zeta_p\notin K$ ?
(2) Suppose $p-1\mid e$ but $p\nmid e$. I am searching a criterion for the congruence $x^{p-1}\equiv-p \pmod{P^{e+1}}$ to be solvable (in $O_K$). Is there “simple” condition that is equivalent to the existence of a solution of such a congruence?
Thanks in advance to anyone who could help me.