On a relative normal integral basis problem over abelian number fields

We say that a Galois extension L/K of a number field K with Galois group G has a relative normal integral basis (RNIB, for short) when the inte- ger ring OL of L is free over the group ring OK [G].Let p be a prime number and assume that K contains a primitive p-th root of unity.In [3], Childs proved that an unramified cyclic extension L/K of degree p has an RNIB if and only if L is obtained by adjoining to K a p-th root of a unit of K satis- fying a certain congruence.Let (K) be the subgroup of K /K consisting of classes [or] (or K ) for which K(cr1/) is unramified over K, and