Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek. "Grothendieck and Witt groups in the reduced theory of quadratic forms." Annales Polonici Mathematici 38.1 (1980): 13-25. <http://eudml.org/doc/263837>.

@article{AndrzejSładek1980,
abstract = {Abstract. Let F be a formally real field. Denote by G(F) and $G_t(F)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G(F)/G_t(F)$. This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G(F)/G_t(F)$, which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the final part of the paper wo apply the results to determine completely the structure of the reduced Grothendieck group $G(F)/G_t(F)$ for all fields with |g(F)| ≤ 16, where g(F) is the factor group F*lT(F), T (F) being the subgroup of all totally positive elements of F. All the results concerning Grothendieck groups have their counter-parts for Witt groups and we also state and prove the results iu that case.},
author = {Andrzej Sładek},
journal = {Annales Polonici Mathematici},
keywords = {Grothendieck group; Witt group; reduced quadratic forms; quasi ordering; formally real fields},
language = {eng},
number = {1},
pages = {13-25},
title = {Grothendieck and Witt groups in the reduced theory of quadratic forms},
url = {http://eudml.org/doc/263837},
volume = {38},
year = {1980},
}

TY - JOUR
AU - Andrzej Sładek
TI - Grothendieck and Witt groups in the reduced theory of quadratic forms
JO - Annales Polonici Mathematici
PY - 1980
VL - 38
IS - 1
SP - 13
EP - 25
AB - Abstract. Let F be a formally real field. Denote by G(F) and $G_t(F)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G(F)/G_t(F)$. This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G(F)/G_t(F)$, which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the final part of the paper wo apply the results to determine completely the structure of the reduced Grothendieck group $G(F)/G_t(F)$ for all fields with |g(F)| ≤ 16, where g(F) is the factor group F*lT(F), T (F) being the subgroup of all totally positive elements of F. All the results concerning Grothendieck groups have their counter-parts for Witt groups and we also state and prove the results iu that case.
LA - eng
KW - Grothendieck group; Witt group; reduced quadratic forms; quasi ordering; formally real fields
UR - http://eudml.org/doc/263837
ER -