Primes of degree one and algebraic cases of Čebotarev's theorem (Q1179588)

Let \(L/K\) be a finite, separable extension of fields, let \(A\subset K\) be a Dedekind domain and let \(B\) be its integral closure in \(L\). The authors prove (Theorem 1) that the ideal class-group of \(B\) is generated by prime ideals which are of degree one over \(A\) and give an example which shows that this may fail for non-separable extensions. A similar assertion holds for ray-class groups in finite extensions of the rationals (Theorem 2). This is applied to give new simple proofs of results of \textit{M. Bauer} [Math. Ann. 77, 353-356 (1916; JFM 46.0249.02)] concerning the determination of a normal extension of \(\mathbb{Q}\) by the set of splitting primes, \textit{M. Deuring} [Klassenkörpertheorie II (Göttingen 1966; Zbl 0247.12004)] dealing with ray class-fields and \textit{J. Wójcik} [Acta Arith. 28, 137-145 (1975; Zbl 0322.12015)], who gave an algebraic proof for some special cases of the Chebotarev density theorem.