On the norm and multiplication principles for norm varieties

Abstract

Let $p$ be a prime, and suppose that $F$ is a field of characteristic zero which is $p$-special (that is, every finite field extension of $F$ has dimension a power of $p$). Let $\alpha \in {\mathcal{𝒦}}_n^M\left(F\right)∕p$ be a nonzero symbol and $X∕F$ a norm variety for $\alpha$. We show that $X$ has a ${\mathcal{𝒦}}_m^M$-norm principle for any $m$, extending the known ${\mathcal{𝒦}}_1^M$-norm principle. As a corollary we get an improved description of the kernel of multiplication by a symbol. We also give a new proof for the norm principle for division algebras over $p$-special fields by proving a decomposition theorem for polynomials over $F$-central division algebras. Finally, for $p=n=m=2$ we show that the known ${\mathcal{𝒦}}_1^M$-multiplication principle cannot be extended to a ${\mathcal{𝒦}}_2^M$-multiplication principle for $X$.