Milnor K-Groups and Finite Field Extensions

Abstract

Let E/F be a finite separable field extension and let m denote the integral part of log2(E : F ). David Leep recently showed that if char(F ) � 2, then for n m the nth power of the fundamental ideal in the Witt ring of E satisfies the equality I n E = I n−m F · I m E. The aim of this note is to prove the analogous equality for the Milnor K-groups, that is KnE = Kn−mF · KmE for n m. In either of these equalities one may not replace m by m − 1, as examples of certain m-quadratic extensions indicate. Mathematics Subject Classifications (2000): 11E81, 12F05, 19D45.