Vol. 76, No. 2, 1978

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When are Witt rings group rings? II

Roger P. Ware

Vol. 76 (1978), No. 2, 541–564

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.

Mathematical Subject Classification 2000
Primary: 12D15
Secondary: 13K05, 10C05
Received: 27 July 1977
Revised: 14 September 1977
Published: 1 June 1978
Roger P. Ware