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arXiv · 2210.03784

The general Arason-Pfister Hauptsatz

Kaique Matias de Andrade Roberto·Hugo Rafael de Oliveira Ribeiro·Hugo Luiz Mariano
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Abstract

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.

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Kaique Matias de Andrade Roberto, Hugo Rafael de Oliveira Ribeiro, Hugo Luiz Mariano. 2024-04-11. The general Arason-Pfister Hauptsatz. https://arxiv.org/abs/2210.03784

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