On the symbol length of symbols

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Fix a prime p and let F be a field with characteristic not p. Let GF be the absolute Galois group of F, and let μps be the GF -module of roots of unity of order dividing ps in a fixed algebraic closure of F. Let α ∈ Hn(F, μ⊗nps) be a symbol (i.e., α=a1 ∪··· ∪ an where ai ∈ H1(F, μps)) with effective exponent dividing ps−1 (that is ps−1α=0 ∈ Hn(GF, μ⊗np). In this work, we show how to write α as a sum of symbols coming from Hn(F, μ⊗n ps−1), that is, symbols of the form pγ for γ ∈Hn(F, μ⊗nps). If n>3 and p=2, we assume F is prime to p closed and of characteristic zero. In the case p=2, we also bound the symbol length of a sum of two symbols with effective exponent dividing ps−1.

Original languageEnglish
Title of host publicationAmitsur Centennial Symposium, 2021
EditorsAvinoam Mann, Louis H. Rowen, David J. Saltman, Aner Shalev, Lance W. Small, Uzi Vishne
PublisherAmerican Mathematical Society
Number of pages13
ISBN (Print)9781470475550
StatePublished - 2024
EventAmitsur Centennial Symposium, 2021 - Jerusalem, Israel
Duration: 1 Nov 20214 Nov 2021

Publication series

NameContemporary Mathematics


ConferenceAmitsur Centennial Symposium, 2021


  • Galois cohomology
  • Milnor K-theory
  • higher symbols
  • quadratic forms

All Science Journal Classification (ASJC) codes

  • General Mathematics


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