Abstract
We study formally real, non-pythagorean fields which have an anisotropic torsion form that contains every anisotropic torsion form as a subform. We obtain consequences for certain invariants and the Witt ring of such fields and construct examples. We obtain a theory analogous to the theory of supreme Pfister forms introduced by Karim Becher and see examples in which the Pythagoras number for formally real fields behaves like the level for nonreal fields.
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