Let $p$ be a prime number, $F$ a number field, and ${\cal H}_F^n$ the set of all unramified cyclic extensions over $F$ of degree $p$ having a relative normal integral basis. When $\z_p \in F^{\times}$, Childs determined the set ${\cal H}_F^n$ in terms of Kummer generators. When $p=3$ and $F$ is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all $p \ge 3$ and $F$ with $\z_p \not\in F^{\times}$ (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.