The general Arason-Pfister Hauptsatz
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929309845815296 |
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| author | Roberto, Kaique Matias de Andrade Ribeiro, Hugo Rafael de Oliveira Mariano, Hugo Luiz |
| author_facet | Roberto, Kaique Matias de Andrade Ribeiro, Hugo Rafael de Oliveira Mariano, Hugo Luiz |
| contents | In the present we develop a fragment of the theory of superfields, polynomials and Marshall's quotient in order to obtain for general special groups, a proof of the Arason-Pfister Hauptsatz (APH): "if $ϕ\neq \emptyset$ is an anisotropic form and $ϕ\in I^n(F)$ then $dim (ϕ) \geq 2^n$". In the process, we also obtain an alternative proof of APH for reduced special groups that avoid the uses of the invariants developed in \cite{dickmann2000special}. The applications of the full Arason-Pfister Hauptsatz leads to interesting properties of graded rings associated to special groups/hyperfields.
\textbf{Keywords:} Arason-Pfister Hauptsatz; hyperfields; special groups; Milnor K-theory; graded rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03784 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The general Arason-Pfister Hauptsatz Roberto, Kaique Matias de Andrade Ribeiro, Hugo Rafael de Oliveira Mariano, Hugo Luiz Commutative Algebra Algebraic Geometry K-Theory and Homology In the present we develop a fragment of the theory of superfields, polynomials and Marshall's quotient in order to obtain for general special groups, a proof of the Arason-Pfister Hauptsatz (APH): "if $ϕ\neq \emptyset$ is an anisotropic form and $ϕ\in I^n(F)$ then $dim (ϕ) \geq 2^n$". In the process, we also obtain an alternative proof of APH for reduced special groups that avoid the uses of the invariants developed in \cite{dickmann2000special}. The applications of the full Arason-Pfister Hauptsatz leads to interesting properties of graded rings associated to special groups/hyperfields. \textbf{Keywords:} Arason-Pfister Hauptsatz; hyperfields; special groups; Milnor K-theory; graded rings. |
| title | The general Arason-Pfister Hauptsatz |
| topic | Commutative Algebra Algebraic Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2210.03784 |