Hilbert 90 for KnM

This chapter formulates a norm-trace relation for the Milnor 𝐾-theory and étale cohomology of a cyclic Galois extension, herein called Hilbert 90 for 𝐾𝑀 𝑛. To begin, the chapter uses condition BL(n) to establish a related exact sequence in Galois cohomology. It then establishes that condition BL(n − 1) implies the particular case of condition H90(n) for 𝓁-special fields 𝑘 such that 𝐾𝑀 𝑛(𝑘) is 𝓁-divisible. This case constitutes the first part of the inductive step in the proof of Theorem A. The remainder of this chapter explains how to reduce the general case to this particular one. The chapter concludes with some background on the Hilbert 90 for 𝐾𝑀 𝑛.