Computing with quadratic forms over 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).