×

zbMATH — the first resource for mathematics

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).

MSC:
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11R58 Arithmetic theory of algebraic function fields
11R99 Algebraic number theory: global fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.