In 1982 T. Craven
generalized the definition of Witt ring from fields to skew fields. The main
aim of this paper is to survey Witt rings of skew fields complete relative to
discrete valuation. We prove that if A is a complete skew field with the
residue class field E of characteristic not 2, then the Witt ring W(A) is
isomorphic to the group ring (W(E)A(Eσ))[Δ], where Δ is the two-element
group, A(Eσ) is the ideal of W(E) generated by all elements of the form
⟨1,−d⟩, d ∈{σ(x)x−1;x ∈ E⋅}E⋅2 and σ is the automorphism of E induced by the
inner automorphism x↦πxπ−1 of A determined by the uniformizer π of A. When A is
commutative, then it turns out to be the well known Springer’s Theorem. The
case of dyadic skew fields is also considered. We show that if A is a finite
dimensional division algebra over a dyadic field, then every binary form over A is
universal.
