## Abstract

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 K^{M} _{n} (F)=p be a nonzero symbol and X/F a norm variety for α. We show that X has a K^{M} _{m} -norm principle for any m, extending the known K^{M} _{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 K^{M} _{1} -multiplication principle cannot be extended to a K^{M} _{2} -multiplication principle for X.

Original language | English |
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Pages (from-to) | 709-720 |

Number of pages | 12 |

Journal | Annals of K-Theory |

Volume | 5 |

Issue number | 4 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Funding Information: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.

Publisher Copyright:

© 2020 Mathematical Sciences Publishers.

## Keywords

- Milnor K-theory
- Norm varieties
- Symbols