On the norm and multiplication principles for norm varieties

Shira Gilat, Eliyahu Matzri

Research output: Contribution to journalArticlepeer-review


Let p be a prime, and suppose that F is a field of characteristic zero which is pspecial (that is, every finite field extension of F has dimension a power of p). Let α ε KM n (F)=p be a nonzero symbol and X/F a norm variety for α. We show that X has a KM m -norm principle for any m, extending the known KM 1 -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 KM 1 -multiplication principle cannot be extended to a KM 2 -multiplication principle for X.

Original languageEnglish
Pages (from-to)709-720
Number of pages12
JournalAnnals of K-Theory
Issue number4
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© 2020 Mathematical Sciences Publishers.


The authors would like to thank Stephen Scully for suggesting the quadratic form used in the counterexample to the higher multiplication principle, and to Stefan Gille for communicating it to us. This research was supported by the Israel Science Foundation (grant no. 630/17). MSC2010: 19D45. Keywords: Milnor K-theory, norm varieties, symbols.

FundersFunder number
Israel Science Foundation19D45, MSC2010, 630/17


    • Milnor K-theory
    • Norm varieties
    • Symbols


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