POWERFUL NECESSARY CONDITIONS FOR CLASS NUMBER PROBLEMS

We give a necessary condition for the ideal class group of a CM-field to be of exponent at most two. This condition enables us to drasticall y reduce the amount of relative class number computation for determina tion of the CM-fields of some types (e.g. the imaginary cyclic non-qua dratic number fields of 2-power degrees) whose ideal class groups are of exponents at most two. We also give a necessary condition for some quartic non-CM-fields to have class number one.