  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.
