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

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 α 𝒦nM(F)p be a nonzero symbol and XF a norm variety for  α. We show that X has a 𝒦mM-norm principle for any m, extending the known 𝒦1M-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 𝒦1M-multiplication principle cannot be extended to a 𝒦2M-multiplication principle for X.

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Milnor $K\mkern-2mu$-theory, norm varieties, symbols
Mathematical Subject Classification 2010
Primary: 19D45
Received: 31 October 2019
Accepted: 12 August 2020
Published: 26 December 2020
Shira Gilat
Department of Mathematics
Bar-Ilan University
Eliyahu Matzri
Department of Mathematics
Bar-Ilan University