It is known that if F is a
superpythagorean field or a nonformally real field with (finite) u-invariant equal to
the number of square classes then the Witt ring of quadratic forms over F is
isomorphic to a group ring Z∕nZ[G] with n = 0,2, or 4 and G a subgroup of the
group of square classes of F. In this paper, we investigate those fields with Witt ring
isomorphic to a group ring Z∕nZ[G] for some n ≧ 0 and some group G.
It is shown that G is necessarily of exponent 2 and such a field is either
superpythagorean or is not formally real with level (Stufe) s(F) = 1 or 2 (so
n = 0,2, or 4). Characterizations of these fields will be given both in terms
of the behavior of their quadratic forms and the structure of their Galois
2-extensions.
