  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).

Number Theory