Computing with quadratic forms over number fields

TL;DR

This paper develops algorithms for analyzing quadratic forms over number fields, including isotropy, hyperbolicity, anisotropic dimension, field invariants, and Witt ring isomorphism, advancing computational number theory.

Contribution

It introduces new algorithms for key properties of quadratic forms over number fields, including isotropy, hyperbolicity, and Witt ring equivalence, with practical computational methods.

Findings

Algorithms for checking isotropy and hyperbolicity over number fields.

Methods for computing the anisotropic part dimension and field invariants.

Algorithm for testing Witt ring isomorphism between number fields.

Abstract

This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number field is either isotropic (respectively locally isotropic) or hyperbolic (respectively locally hyperbolic). Next we give a method of computing the dimension of an anisotropic part of a quadratic forms. The second part of the paper is devoted to algorithms computing two field invariants: the level and the Pythagoras number. Ultimately we present an algorithm verifying whether two number fields have isomorphic Witt rings (i.e. are Witt equivalent).