Decomposition of Witt rings and Galois groups

TO each field F of characteristic not 2, one can associate a certain Galois group G(F), the so-called W-group off, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of G(F). In an appendix we provide a complete list of Witt rings and corresponding W-groups for fields F with \Fover dot/Fover dot(2);2 less than or equal to 16.