#### Vol. 5, No. 4, 2020

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On the norm and multiplication principles for norm varieties

### Shira Gilat and Eliyahu Matzri

Vol. 5 (2020), No. 4, 709–720
##### 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 $\alpha \in {\mathsc{𝒦}}_{n}^{M}\left(F\right)∕p$ be a nonzero symbol and $X∕F$ a norm variety for  $\alpha$. We show that $X$ has a ${\mathsc{𝒦}}_{m}^{M}$-norm principle for any $m$, extending the known ${\mathsc{𝒦}}_{1}^{M}$-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 ${\mathsc{𝒦}}_{1}^{M}$-multiplication principle cannot be extended to a ${\mathsc{𝒦}}_{2}^{M}$-multiplication principle for $X$.

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##### Keywords
Milnor $K\mkern-2mu$-theory, norm varieties, symbols
Primary: 19D45
##### Milestones
Accepted: 12 August 2020
Published: 26 December 2020
##### Authors
 Shira Gilat Department of Mathematics Bar-Ilan University Ramat-Gan Israel Eliyahu Matzri Department of Mathematics Bar-Ilan University Ramat-Gan Israel