You can prove that the class number of a cyclotomic number field of an odd prime order is divisible by that of its subfield by using class field theory. Since it has the unique quadratic subfield and its class number can be relatively easily computed when the discriminant is small, you can get useful information of the class number of the cyclotomic number field.
For example, you can find the proof here: http://math.stackexchange.com/questions/175718/on-the-class-number-of-a-cyclotomic-number-field-of-an-odd-prime-orderhttps://math.stackexchange.com/questions/175718/on-the-class-number-of-a-cyclotomic-number-field-of-an-odd-prime-order
