The general Arason-Pfister Hauptsatz

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.