Generators of the Witt groups of algebraic integers
Abstract
For a number field K let 𝓞K be the ring of algebraic integers of K. A basic result on the Witt ring W𝓞K of symmetric bilinear forms over the ring 𝓞K was established in [MH]. The structure of the Witt group W𝓞K, 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 W𝓞K into direct sum of cyclic groups. We will also describe products of these generators. This completely determines the structure of the ring W𝓞K. As an illustration of these results we determine the structure of Witt rings W𝓞K 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 W𝓞K → W𝓞L (for details see [Cz2]).
Keywords
Witt ring; ring of algebraic integers
References
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8. P. Shastri, Witt groups of algebraic integers, J. Number Theory, 30 (1988), 243-266.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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