The general Arason-Pfister Hauptsatz

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 $\phi \neq \emptyset$ is an anisotropic form and $\phi \in I^n(F)$ then $dim (\phi) \geq 2^n$". 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.