Consider an algebraic number $\alpha$, which can be taken to be an integer. With $\deg\alpha$ a prime number, one can easily arrange that to be such that all powers $\alpha^n$ to be of the same degree as $\alpha$.

I would like to know how to get numbers with the same property without any restriction on its degree. If $\deg\alpha=n$, my guess for a condition to guarantee this behaviour is $\mathbf{Q}[\alpha]$ be linearly disjoint with the $n$-th cyclotomic extension.

My second question is: Is it possible for a number field to be linearly disjoint with every cyclotomic extension?