Abstract
Rigid elements in a field, should they exist, have strong influence on structure of the Witt ring of the field. We generalize rigid elements to the context of tt-fold Pfister forms in two ways and study the relations between n-rigid and super n-rigid elements in various classes of fields including global fields, and in abstract Witt rings. In special cases the existence of higher rigidities turns out to be equivalent to important properties of fields known in literature. Among these are the property An9 torsion freeness of ΓF and finite stability index of WF.
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