Artin's \(L\)-functions and Gassmann equivalence (Q1086622)

As is well known, two algebraic number fields \(K\) and \(K'\) have the same zeta-function if and only if the prime numbers have the same type of decomposition in these fields (or are ramified), i.e. if the fields \(K\) and \(K'\) are 'arithmetically equivalent' over \(\mathbb Q\). If the ground field is not \(\mathbb Q\), this does not hold any longer, but one can characterize arithmetical equivalence as follows: Take some Galois extensions \(N/k\) containing \(K\) and \(K'\) and let \(G\) be its Galois group. Then \(G\) acts on the conjugates of \(K\) respectively of \(K'\); and these permutation representations of \(G\) have the same character if and only if \(K\) and \(K'\) are arithmetical equivalent over \(k\). Apparently not knowing the reviewer's paper on this topic [J. Reine Angew. Math. 299-300, 342--384 (1978; Zbl 0367.12003)] the author reproves this and related results and characterizes arithmetical equivalence by one new condition, stating that some Artin \(L\)-functions connected with these permutation characters coincide.