The anisotropic part of a quadratic form over a number field

TL;DR

This paper introduces an algorithm for explicitly constructing the anisotropic component of a quadratic form over any number field, enhancing understanding and manipulation of such forms in algebraic number theory.

Contribution

It provides a novel algorithm to compute the anisotropic part of quadratic forms over arbitrary number fields, which was previously not explicitly available.

Findings

Algorithm successfully constructs anisotropic parts over various number fields

Enhances computational tools in algebraic number theory

Facilitates further research on quadratic forms and their classifications

Abstract

It is well known that every non-degenerate quadratic form admits a decomposition into an orthogonal sum of its anisotropic part and a hyperbolic form. This decomposition is unique up to isometry. In this paper we present an algorithm for constructing an anisotropic part of a given form with coefficients in an arbitrary number field.