The general Arason-Pfister Hauptsatz
Kaique Matias de Andrade Roberto, Hugo Rafael de Oliveira Ribeiro, and Hugo Luiz Mariano

TL;DR
This paper develops a fragment of superfield theory to prove the Arason-Pfister Hauptsatz for general special groups, providing new proofs and exploring implications for graded rings and hyperfields.
Contribution
It introduces a new fragment of superfield theory to prove the Arason-Pfister Hauptsatz for general special groups and offers an alternative proof for reduced special groups.
Findings
Proof of APH for general special groups
Alternative proof for reduced special groups
Properties of graded rings related to special groups
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 is an anisotropic form and then ". 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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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
