AbstractWe 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 drastically reduce the amount of relative class number computation for determination of the CM ‐ fields of some types (e. g. the imaginary cyclic non ‐quadratic 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