  Let $U/L$ be a finite abelian extension of number fields. We first construct
a universal primitive generator of $U$ over $L$ whose relative trace to any
intermediate field $F$ becomes a generator of $F$ over $L$, too. We also
develop a similar argument in terms of norm. As its examples we investigate
towers of ray class fields over imaginary quadratic fields. And, we further
present a new method of finding a normal element for the extension $U/L$.
