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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  shows that k A ^k' A implies ζ*(s)-ζ Λ /(s)doi:10.2996/kmj/1138847384 fatcat:c36x6poicjhldnpzk2ebeptmli