For a number field K let ????K be the ring of algebraic integers of K. A basic result on the Witt ring W????K of symmetric bilinear forms over the ring ????K was established in [MH]. The structure of the Witt group W????K, in terms of arithmetical invariants of K, was determined in [Sh]. Here we state precisely this description. We find generators of cyclic direct summands in the decomposition of the group W????K into direct sum of cyclic groups. We will also describe products of these generators. This completely determines the structure of the ring W????K. As an illustration of these results we determine the structure of Witt rings W????K for all quadratic, and some cubic and some biquadratic fields K. The results of this paper allow us to find arithmetical conditions for the existence of an isomorphism of Witt rings W????K → W????L (for details see [Cz2]).
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Vol. 12 (1998)
Published: 1998-09-30
10.1515/amsil
English