A Harris and Segal theorem for exceptional rings of algebraic integers

Let K be a number field, O-K the ring of the integers of K, l a prime integ er and Z((l)) the localisation of Z at l. Hart-is and Segal [4] proved that there exists infinitely many primes p of O-K such that the natural morphis m K-i(O-K) x Z((l)) --> K-i (O-K/p) x Z((l)) in algebraic K-theory is split surjective for i > 0, except if l = 2 and K is exceptional. In this Note, we prove that the Harris-Segal theorem is still true for l = 2 in the excep tional case, if we replace algebraic K-theory by orthogonal K-theory define d by Karoubi [5]. Thanks to [3], we can then determine a direct summand of the 2-torsion of KOn(O-K). (C) Academie des Sciences/Elsevier, Paris.