Remark on the divisibility of the class numbers of certain quartic number fields by 5

The congruence relation modulo 5 between the class numbers of the real and imaginary quartic subfields of the extension of a quadratic number field obtained by adjoining a fifth root of unity is studied. 1. Introduction. The aim of this note is to study the congruence relation modulo 5 between the class numbers of the real and imaginary quartic subfields of the extension of a quadratic number field obtained by adjoining a fifth root of unity. Our result gives different proofs of the main theorems of Parry (6, 7). Let m be a square free rational integer prime to 5, f a primitive fifth root of unity and Q the field of rational numbers. Let K = Q({m , f)> which is of degree 8 over Q and has the following subfields