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The goal of this paper is to present proofs of two results of Markus Rost, the Chain Lemma 1 and the Norm Principle 3. These are the steps needed to complete the published verification of the Bloch– Kato conjecture, that the norm residue maps are isomorphisms for every prime p, every n and every field k containing 1/ p.

Norm Varieties and the Chain Lemma (After Markus Rost)