Existence of Rost Varieties

Abstract

This chapter proves that the norm varieties constructed in the previous chapter are indeed Rost varieties. In other words, it proves that Rost varieties exist. In doing so, the chapter also proves the Norm Principle, which is a theorem that supposes that 𝑘 is an 𝓁-special field of characteristic 0, and that 𝑋 is a norm variety for some nontrivial symbol ª. Then each element of ̅𝐻−1, −1(𝑋) is a Kummer element. In preparation for the proof of the Norm Principle, this chapter develops some basic facts about elements of ̅𝐻−1, −1(𝑋) supported on points 𝑥 with 𝑘(𝑥) : 𝑘=𝓁.