Let $K$ be a field, and $\alpha$ be an element of a separable extension of $K$, such that $\alpha^p\in K$, but $\alpha\notin K$, $p$ a prime.
Let $f$ be the minimal polynomial of $\alpha$ over $K$ and $\beta$ be another root of $f$.
Question 1. Is it true that $\alpha$ and $\beta$ differ by a $p$-th root of unity? i.e. if $\epsilon=\alpha/\beta$, the is $\epsilon^p=1$?
Question 2. In general, is it true that any two roots of an irreducible polynomial over a field always differ (multiplicatively) by a root of unity?
Reference: Galois Theory - Ian Stewart, Third Ed., p. 156.