Decomposability of orthogonal involutions in degree 12
| Autor: | Anne Quéguiner-Mathieu, Jean-Pierre Tignol |
|---|---|
| Přispěvatelé: | UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Quaternion algebra General Mathematics 010102 general mathematics Primary 11E72 Secondary 16W10 11E81 K-Theory and Homology (math.KT) Field (mathematics) 01 natural sciences Tensor product Discriminant Quadratic form Mathematics - K-Theory and Homology 0103 physical sciences FOS: Mathematics Binary quadratic form 010307 mathematical physics 0101 mathematics Invariant (mathematics) Central simple algebra Mathematics |
| Zdroj: | Pacific Journal of Mathematics, Vol. 304, no.1, p. 169-180 (2020) |
| Popis: | A theorem of Pfister asserts that every $12$-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from $2$ decomposes as a tensor product of a binary quadratic form and a $6$-dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions: every central simple algebra of degree $12$ with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree $6$ with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree $12$, and to calculate the $f_3$-invariant of the involution if the algebra has index $2$. |
| Databáze: | OpenAIRE |
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