Model-theoretic Results for Pythagorean Fields

Max Dickmann & Francisco Miraglia

Notation. (a) A field is Pythagorean if every sum of squares is a square. All Pythagorean fields considered here are formally real (= orderable).

(b) For any field F and n ≥ 1, I^{n}(F) denotes the n-th power of the ideal of forms of even dimension in the Witt ring of quadratic forms over a field F (modulo Witt-equivalence).

(c) We consider (formally real) fields endowed with their \underline{special group} (rather than field) language L_{SG}, where addition is replaced by the (much weaker) binary relation

a ∈ D_{F} (1, b) iff F ⊧ (∃x y ∈ ∑ F^{2})(a = x + by)

(a, b ∈ F^{×}). The motivation for, and the utility of considering this language can be found in [1]. G_{red}(F) denotes the (multiplicative) group F^{×}/ ∑ F^{×}^{2} endowed with this relation and the constant −1/ ∑ F^{×}^{2}. Theorem 1. (Bounds in the theory of Pythagorean fields)

(a) There exists a recursive function f : N ×N ⟶ N such that for every n, m ≥ 1, any Pythagorean field F, and any regular quadratic form φ of dimension m over F, if φ ∈ I^{n}(F), then φ is a linear combination of at most f(n, m) Pfister forms of degree n. It follows that membership of I^{n}(F) can be expressed, uniformly on dimension, by a first-order formula of the language L_{SG} above.

(b) The bound f(n, m) in (a) is simply exponential, of the form m^{n} with a suitable coefficient, for a very extensive class of Pythagorean fields F (including, f. ex., all those whose associated special group G(F) is locally finite).

Item (a) follows by a compactness argument from our solution to Marshall’s problem in quadratic form theory ([2]). Item (b) requires a (rather involved) combinatorial argument. Theorem 2. (First-order properties of the Pythagorean closure)

(a) For any formally real field F, the special group embedding G_{red}(F) ↪ G(F^{p}) induced by the inclusion of F into its Pythagorean closure F^{p} is not elementary (for the language L_{SG}) whenever the space of orders of F is not perfect.

(b) There are fields F for which the embedding in (a) is not pure. However,

(c) The morphism in (a) reflects isotropy for all formally real fields F.

These results follow from our Boolean-theoretic methods expounded in [1].

[1] M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, 247 pp., Memoirs of the AMS 689, May 2000.

[2] M. Dickmann, F. Miraglia, On forms whose total signature is zero mod 2^{n}. Solution to a problem of M. Marshall, Inventiones Math. 133, 243 - 278 (1998).

Addresses Max Dickmann : Equipe de Logique Mathématique (UPRESA 7056), 2, Place Jussieu, Case 7012, 75251 Paris Cedex 05, FRANCE

E-mail: dickmann@logique.jussieu.fr

Francisco Miraglia: Instituto de Matemática e Estatística, Universidade de São Paulo, C. P. 66.281 (Ag. Cidade São Paulo), 05389-970 S.Paulo, S.P., BRASIL

E-mail: miraglia@ime.usp.br