Let Q be the rational number field, Q the algebraic closure of Q and k (kaQ) an algebraic number field of finite degree. Let ζ k (s) be the Dedekind zeta-function of k, k A the adele ring of k and G k the Galois group of Q/k with Krull topology. We adopt similar notations for an algebraic number field k f (k f d.Q} of finite degree. If the extension k/Q is a finite Galois extension and if ζ Λ (s)=ζ Λ ,(s), then k=k' (cf. Lemma 2). The Lemma 7 of [3] shows that k A ^k' A implies ζ*(s)-ζ Λ /(s)

more »

... f. Corollary of Lemma 3). We also proved that G k ^Gk > implies ζ k (s)=ζk'(s) (cf. [6] or [4]). From the above results, it is natural and interesting to consider whether, for any algebraic number fields k and k' of finite degree, k A ^kA implies k = k f and whether G k ^Gk , implies k^k f . In Theorem 1, we shall show that there exist algebraic number fields k and k f of finite degree satisfying the following conditions :