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


Zbl 0807.11024
Full Text: DOI


[1] Baeza R., J. Algebra 92 pp 446– (1985) · Zbl 0553.10016 · doi:10.1016/0021-8693(85)90133-4
[2] Carpenter J., Thesis (1989)
[3] Czogala A., Thesis (1987)
[4] Czogala A., Acta Arith 58 (1987)
[5] Endler O., Abh. Math. Sem. Hamburg 33 pp 80– (1969) · Zbl 0169.36602 · doi:10.1007/BF02992808
[6] Hasse H., Zahlentheorie (1963)
[7] Lam T.Y., The algebraic theory of quadratic forms (1980) · Zbl 0437.10006
[8] Marshall M., Queen’s Papers in Pure and Appl. Math 57 (1980)
[9] Milnor J., Symmetric oilinear forms (1973) · doi:10.1007/978-3-642-88330-9
[10] OMeara O.T., Introduction to quadratic forms (1971)
[11] Perlis R., Matching Witts with global fields (1989) · Zbl 0807.11024
[12] Szyrniczek K., Math. Slovaca (1989)
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