Class number relations between pure fields and their Galois closures

Let p be an odd prime and let m be a positive p-th power free integer greater than one. Let K be an algebraic number field generated by the positive p-th root of m over the field of rational numbers, L its Galois closure in the field of complex numbers, and k the p-th cyclotomic field. We calculate a class number relation among them, which gives a sufficient condition for the class number of K to be a multiple of p. Moreover, we show that a rational prime q unequal to p is a divisor of the class number of L if and only if q divides that of k or K.