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Witt equivalence of global fields. (English) Zbl 0724.11020
The author continues the work of his joint paper with P. E. Conner, R. Litherland and R. Perlis [Matching Witts with global fields, preprint 1989, Contemp. Math. 155, 365–387 (1994; Zbl 0807.11024)]. In §1 he puts up a list of (finitely many) integral invariants of a global field \(F\) which determine the Witt ring \(W(F)\) of symmetric bilinear forms over \(F\) up to ring isomorphism. The function field case is nearly trivial. In §2 he derives 8 conditions between these invariants which are necessary and sufficient for the existence of a number field \(F\) with the given invariants. This implies in particular that the number w(n) of different Witt rings of number fields of absolute degree n is finite and explicitly computable by means of certain partition functions \(p(n)\), \(p_ 1(n)\). In §3 he gives complete lists and decision procedures for \(n=2,3\). There are 7 resp. 8 different isomorphism classes for \(W(F)\). The proofs are clear and elementary (modulo number theory).

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11R58 Arithmetic theory of algebraic function fields
11R99 Algebraic number theory: global fields
Full Text: DOI
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