# Grothendieck and Witt groups in the reduced theory of quadratic forms

Annales Polonici Mathematici (1980)

- Volume: 38, Issue: 1, page 13-25
- ISSN: 0066-2216

## Access Full Article

top## Abstract

top## How to cite

topAndrzej 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.