Linkage of symbol p-algebras of degree 3

  • Adam ChapmanEmail author


Given a field F of characteristic 3 and division symbol p-algebras \([\alpha ,\beta )_{3,F}\) and \([\alpha ,\gamma )_{3,F}\) of degree 3 over F, we prove that if \(\alpha {{\text {dlog}}}(\beta )\wedge {{\text {dlog}}}(\gamma )\) is trivial in the Kato–Milne cohomology group \(H_3^3(F)\), then the algebras share a common splitting field which is an inseparable degree 3 extension of either F or a quadratic extension of F. In the special case of quadratically closed fields, if \(\alpha {{\text {dlog}}}(\beta )\wedge {{\text {dlog}}}(\gamma )=0\), then they share an inseparable degree 3 extension of F.


Kato–Milne cohomology Fields of positive characteristic Central simple algebras Division algebras Symbol algebras p-algebras Linkage 

Mathematics Subject Classification

Primary 16K20 Secondary 11E04 11E81 19D45 



The author thanks the anonymous referee for the careful reading of the submitted manuscript and the useful comments.


  1. 1.
    Chapman, A.: Common subfields of \(p\)-algebras of prime degree. Bull. Belg. Math. Soc. Simon Stevin 22(4), 683–686 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chapman, A., Dolphin, A.: Differential forms, linked fields, and the \(u\)-invariant. Arch. Math. (Basel) 109(2), 133–142 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chapman, A., Dolphin, A.: Types of linkage of quadratic Pfister forms. J. Number Theory 199, 352–362 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chapman, A., Dolphin, A., Laghribi, A.: Total linkage of quaternion algebras and Pfister forms in characteristic two. J. Pure Appl. Algebra 220(11), 3676–3691 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chapman, A., Dolphin, A., Leep, D.B.: Triple linkage of quadratic Pfister forms. Manuscr. Math. 157(3–4), 435–443 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chapman, A., Gilat, S., Vishne, U.: Linkage of quadratic Pfister forms. Comm. Algebra 45(12), 5212–5226 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chapman, A., McKinnie, K.: The \(u^n\)-invariant and the symbol length of \(H_2^n(F)\). Proc. Am. Math. Soc. 147(2), 513–521 (2019)CrossRefGoogle Scholar
  8. 8.
    Elduque, A., Villa, O.: A note on the linkage of Hurwitz algebras. Manuscr. Math. 117(1), 105–110 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gille, P.: Invariants cohomologiques de Rost en caractéristique positive. \(K\)-Theory 21(1), 57–100 (2000)Google Scholar
  10. 10.
    Gille, P., Szamuely, T.: Central Simple Algebras and Galois cohomology, Volume 101 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2006)Google Scholar
  11. 11.
    Morandi, P.: The Henselization of a valued division algebra. J. Algebra 122(1), 232–243 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Saltman, D.J.: Lectures on Division Algebras, Volume 94 of CBMS Regional Conference Series in Mathematics. Published by American Mathematical Society, Providence, RI; on behalf of Conference Board of the Mathematical Sciences, Washington, DC (1999)Google Scholar
  13. 13.
    Tignol, J.-P., Wadsworth, A.R.: Value Functions on Simple Algebras, and Associated Graded Rings. Springer Monographs in Mathematics. Springer, Berlin (2015)zbMATHGoogle Scholar
  14. 14.
    Tikhonov, S.V.: Division algebras of prime degree with infinite genus. Tr. Mat. Inst. Steklova 292(Algebra, Geometriya i Teoriya Chisel), 264–267 (2016). English version published in Proc. Steklov Inst. Math. 292 (2016), no. 1, 256–259MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceTel-Hai Academic CollegeUpper GalileeIsrael

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