Relative integral bases over a Hilbert class field Original Research Article

Let K be a number field, H its Hilbert class field and L a Galois extension of K containing H. In this paper, we prove that LH has a relative integral basis (RIB) if the order of G=Gal(LH) is odd or if the 2-Sylow subgroups of G are not cyclic. If the order of G is even and the 2-Sylow subgroups are cyclic we reduce the problem of the existence of a RIB to a quadratic extension of H.