Generalized Steinberg relations

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We consider a field F and positive integers n, m, such that m is not divisible by Char (F) and is prime to n!. The absolute Galois group GF acts on the group Un(Z/ m) of all (n+ 1) × (n+ 1) unipotent upper-triangular matrices over Z/ m cyclotomically. Given 0 , 1 ≠ z∈ F and an arbitrary list w of n Kummer elements (z) F, (1 - z) F in H1(GF, μm) , we construct in a canonical way a quotient Uw of Un(Z/ m) and a cohomology element ρz in H1(GF, Uw) whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case n= 2 recovers the Steinberg relation in Galois cohomology, proved by Tate.

Original languageEnglish
Article number92
JournalResearch in Number Theory
Issue number4
StatePublished - 17 Oct 2022


  • Galois cohomology
  • Kummer map
  • Massey products
  • Steinberg relations

ASJC Scopus subject areas

  • Algebra and Number Theory


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