TY - JOUR

T1 - Common splitting fields of symbol algebras

AU - Chapman, Adam

AU - Florence, Mathieu

AU - McKinnie, Kelly

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/7

Y1 - 2023/7

N2 - We study the common splitting fields of symbol algebras of degree pm over fields F of char (F) = p. We first show that if any finite number of such algebras share a degree pm simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees pm0,⋯,pmt share a cyclic splitting field of degree pm0+⋯+mt. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol’s to bound the symbol length of classes in Brpm(F) whose symbol length when embedded into Brpm+1(F) is 2 for p∈ { 2 , 3 }. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

AB - We study the common splitting fields of symbol algebras of degree pm over fields F of char (F) = p. We first show that if any finite number of such algebras share a degree pm simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees pm0,⋯,pmt share a cyclic splitting field of degree pm0+⋯+mt. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol’s to bound the symbol length of classes in Brpm(F) whose symbol length when embedded into Brpm+1(F) is 2 for p∈ { 2 , 3 }. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

UR - http://www.scopus.com/inward/record.url?scp=85131602600&partnerID=8YFLogxK

U2 - 10.1007/s00229-022-01401-2

DO - 10.1007/s00229-022-01401-2

M3 - Article

AN - SCOPUS:85131602600

SN - 0025-2611

VL - 171

SP - 649

EP - 662

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 3-4

ER -