1. It was proved by E. Artin 1 that if k is an algebraic number field (of finite degree) and K a normal extension field with the icosahedral group as the Galois group with regard to k, then the zeta-function g (s, kc) of k divides the zeta-function g(s, K), in the sense that the quotient t (s, K) /g (s, kc) is an integral function of the complex variable s. Using Artin's method, we shall show in this note that the zeta-function t(s, k) of an algebraic number field kc divides the zeta-function g(s, K) of every normal extension field K of 7c.1' The proof (3) is based on a group-theoretical lemma which is established in 2. Our result will enable us to prove in 4 the following theorem conjectured by C. L. Siegel.2 Consider all algebraic number fields of a fixed degree n. Let d be the discriminant of 7k, let h be the number of classes of ideals in k, and let ] be the regulator of 7k. Then