Karim Johannes Becher

Milnor K-Groups and Finite Field Extensions

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.

The Elman-Lam-Krüskemper Theorem

For a (formally) real field 𝐾, the vanishing of a certain power of the fundamental ideal in the Witt ring of √𝐾(−1) implies that the same power of the fundamental ideal in the Witt ring of 𝐾 is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of 𝐾. This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of 𝐾. We present an exposition of these results.