×

Matching Witts locally and globally. (English) Zbl 0766.11023

Let \(F\) and \(E\) be global fields of characteristic not 2, let \(W(F)\) and \(W(E)\) be their Witt rings of nondegenerate symmetric bilinear forms, and let \(\Omega(F)\) and \(\Omega(E)\) be their respective sets of primes (including infinite primes, if any). The fields \(F\) and \(E\) are said to be Witt equivalent if \(W(F)\) and \(W(E)\) are isomorphic. On the other hand, they are said to be reciprocity equivalent if there exist a bijective map \(T: \Omega(F)\to \Omega(E)\) and a group isomorphism \(t: \dot F/\dot F^ 2 \to \dot E/\dot E^ 2\) preserving Hilbert symbols; i.e., \((a,b)_ P=(ta,tb)_{TP}\) for all \(P\in\Omega(F)\) and all \(a,b\in \dot F/\dot F^ 2\). The fact that reciprocity equivalence implies Witt equivalence follows from the Hasse principle and a criterion for Witt equivalence involving the represented value sets of binary quadratic forms. Moreover, the existence of a reciprocity equivalence between \(F\) and \(E\) is equivalent to the existence of isomorphisms between the local Witt rings \(W(F_ P)\) and \(W(E_{TP})\) for all \(P\in\Omega(F)\).
The main purpose of the paper under review is to prove that, in fact, two global fields are Witt equivalent if and only if they are reciprocity equivalent. The proof that Witt equivalence implies reciprocity equivalence proceeds in two stages. First, valuation theory and rigidity properties of the fields are used to prove that the fields are what the author calls almost reciprocity equivalent, the definition for which is obtained by changing the sets \(\Omega(F)\) and \(\Omega(E)\) in the definition of reciprocity equivalence to the sets \(\Omega_ 1(F)\) and \(\Omega_ 1(E)\) consisting of finite non-dyadic primes and the infinite real primes of the respective fields. Then, by arguments attributed by the author to R. Perlis, it is shown that an almost reciprocity equivalence of two Witt equivalent fields can be extended to a reciprocity equivalence.
A consequence of the main theorem, and the elementary fact that reciprocity equivalence preserves degree over \(\mathbb{Q}\), is that Witt equivalent algebraic number fields have the same degree over \(\mathbb{Q}\). The main result of this paper has also been proved using different methods based on an analysis of the 2-torsion in the Brauer group by R. Perlis, the author, P. E. Conner and R. Litherland [“Matching Witts with global fields”, preprint, 1989; per ibid.].

MSC:

11E12 Quadratic forms over global rings and fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] ARASON J. K., ELMAN R., JACOB B.: Rigid elements, valuations and realization of Witt rings. J. Algebra, 110 (1987), 449-467. · Zbl 0629.10016
[2] BAEZA R., MORESI R.: On the Witt equivalence of fields of characteristic 2. J. Algebra, 92 (1985), 446-453. · Zbl 0553.10016
[3] CARPENTER J.: Finiteness Theorems for Forms over Number Fields. Dissertation, LSU Baton Rouge, La., 1989.
[4] CZOGALA A.: Witt Rings of Algebraic Number Fields. (Polish). Dissertation, Silesian University, Katowice, 1987.
[5] CZOGALA A.: On reciprocity equivalence of quadratic number fields. Acta Arith. · Zbl 0733.11012
[6] HARRISON D. K.: Witt Rings. Univ. of Kentucky, 1970.
[7] LAM T. Y.: The Algebraic Theory of Quadratic Forms. Benjamin/Cummings, Reading, Mass, 1980. · Zbl 0437.10006
[8] MILNOR J., HUSEMOLLER D.: Symmetric Bilinear Forms. Springer Verlag, 1973. · Zbl 0292.10016
[9] O’MEARA O.T.: Introduction to Quadratic Forms. Springer Verlag, 1971. · Zbl 0207.05304
[10] PALFREY T.: Density Theorems for Reciprocity Equivalence. Dissertation, LSU Baton Rouge, La., 1989.
[11] PERLIS R., SZYMICZEK K., CONNER P. E., LITHERLAND R.: Matching Witts with global fields. Preprint (1989). · Zbl 0807.11024
[12] SZYMICZEK K.: Problem No. 7. Proc. Summer School on Number Theory, Chlébské 1983. J. E. Purkyně Univ., Brno, 1985.
[13] SZYMICZEK K.: Witt equivalence of global fields. Commun. Algebra 19 (1991), 1125-1149. · Zbl 0724.11020
[14] WARE R.: Valuation rings and rigid elements in fields. Canad. J. Math., 33 (1981), 1338-1355. · Zbl 0514.10015
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.