On the symbol length of symbols

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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
Pages219-231
Number of pages13
ISBN (Print)9781470475550
StatePublished - 2024
EventAmitsur Centennial Symposium, 2021 - Jerusalem, Israel
Duration: 1 Nov 20214 Nov 2021

Publication series

NameContemporary Mathematics
Volume800

Conference

ConferenceAmitsur Centennial Symposium, 2021
Country/TerritoryIsrael
CityJerusalem
Period1/11/214/11/21

Keywords

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

All Science Journal Classification (ASJC) codes

  • General Mathematics

Fingerprint

Dive into the research topics of 'On the symbol length of symbols'. Together they form a unique fingerprint.

Cite this