Let $K$ be a number field of degree $n$ over the rationals. Under what conditions does there exist a non-rational algebraic integer $\alpha $ in $K$ such that the discriminant of $\alpha $ divides the norm of $\alpha$? 

This question was first asked on Math StackExchange, Question [2923849][1], two weeks ago. 


  [1]: https://math.stackexchange.com/questions/2923849/