A remark on Dedekind zeta functions and the \(K\)-group (Q923623)

Assume \(k\) and \(k'\) are two number fields having the same Dedekind zeta function. Let \(L\) be a (common) Galois-closure and \(p\) a prime not dividing \([L:k]\). The author shows that for all \(n\geq 0\) the \(p\)-primary components of the \(K\)-groups of the rings of integers of \(k\) and \(k'\) are isomorphic. The result is easily obtained from the case \(n=0\), which is due to \textit{R. Perlis} [J. Number Theory 10, 489--509 (1978; Zbl 0393.12009)].