Quadratic forms, local-global principles, and field invariants

The Hasse-Minkowski Theorem states that a quadratic form defined over a global field is isotropic if and only if it is isotropic over all completions of the field, and is one of the first examples of a local-global principle for quadratic forms. In this thesis, we investigate local-global principles for quadratic forms over more general fields and their use in answering several questions about quadratic forms. First, we study the validity of the local-global principles for isotropy and isometry of quadratic forms over finitely generated field extensions with respect to various sets of discrete valuations. Next, we use the local-global principle for isotropy to study anisotropic universal quadratic forms, particularly over semi-global fields. Finally, we use the Witt index to ask refined questions about the local-global principle for isotropy and about universal quadratic forms.