Witt rings of quadratically presentable fields

Document Type : Research Paper


1 Institute of Mathematics, Faculty of Mathematics, Physics and Chemistry, University of Silesia

2 Laboratorire de Math'{e}matiques, Universit'{e} Savoie Mont Blanc, B^{a}timent Le Chablais, Campus Scientifique, 73376 Le Bourget du Lac, France.



This paper introduces an approach to the axiomatic theory of quadratic forms based on $presentable$ partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of $quadratically\  presentable\  fields$, that is, fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. In particular, Witt rings of symmetric bilinear forms over fields of arbitrary characteristics are isomorphic to Witt rings of suitably built quadratically presentable fields.


[1] Carson, A.B. and Marshall, M., Decomposition of Witt rings, Canad. J. Math. 34 (1982), 1276-1302.
[2] Cassels, J.W.S., On the representation of rational functions as sums of squares, Acta Arith. 9 (1964), 79-82.
[3] Cordes, C., The Witt group and equivalence of fields with respect to quadratic forms, J. Algebra 26 (1973), 400-421.
[4] Cordes, C., Quadratic forms over non-formally real fields with a finite number of quaternion algebras, Pacific J. Math. 63 (1973), 357-365.
[5] Dickmann, M., Anneaux de Witt abstraits et groupes speciaux, In: Seminaire de structures algebriques ordonnees 42, University of Paris 7, 1993.
[6] Dickmann, M. and Miraglia, F., "Special groups: Boolean-theoretic methods in the theory of quadratic forms", Mem. Amer. Math. Soc. 145(689), 2000, http://dx.doi.org/10.1090/memo/0689.
[7] Dickmann, M. and Petrovich, A., Real semigroups and abstract real spectra I, Contemp. Math. 344 (2004), 99-119.
[8] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., "A Compendium of Continuous Lattices", Springer, 1980.
[9] Gładki, P. and Marshall, M., Witt equivalence of function fields over global fields, Trans. Amer. Math. Soc. 369 (2017), 7861-7881.
[10] Kleinstein, J.L. and Rosenberg, A., Succinct and representational Witt rings, Pacific J. Math. 86 (1980), 99-137.
[11] Knebusch, M., Rosenberg, A., and Ware, R., Structure of Witt rings, quotients of abelian group rings and orderings of fields, Bull. Amer. Math. Soc. (N.S.) 77 (1971), 205-210.
[12] Knebusch, M., Rosenberg, A., and Ware, R., Structure of Witt rings and quotients of Abelian group rings, Amer. J. Math. 94 (1972), 119-155.
[13] Kula, M., Szczepanik, L., and Szymiczek, K., Quadratic form schemes and quaternionic schemes, Fund. Math. 130 (1988), 181-190.
[14] Lam, T.Y., "Introduction to Quadratic Forms over Fields", Graduate Studies in Mathematics 67, Amer. Math. Soc., 2005.
[15] Marshall, M., Abstract Witt rings, Queen’s Pap. Pure Appl. Math. 57, Queen’s University, 1980.
[16] Marshall, M., Real reduced multirings and multifields, J. Pure Appl. Algebra 205 (2006), 452-468.
[17] Pfister, A., Quadratische Formen in beliebigen Körpern, Invent. Math. 1(2) (1966), 116-132.
[18] Szczepanik, L., Fields and quadratic form schemes with the index of radical not exceeding 16, Ann. Math. Sil. 1(13) (1985), 23-46.
[19] Szczepanik, L., Quadratic form schemes with non-trivial radical, Colloq. Math. 2(49) (1985), 143-160.
[20] Witt, E., Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44.