Classification of Hermitian forms over central simple algebras with involution

Let W (A, σ) be a central simple algebra with involution over a field F. In 1998 Bayer-Fluckiger and Parimala claimed that if I3 F is torsion-free, then hermitian forms over (A, σ) are classified up to isometry by invariants.{09}In this thesis, it is shown that their hypothesis is insufficient for classification and that I 3 F($-1$) = 0 naturally gives their results. Furthermore, as I 3 F($-1$) = 0 implies a reduced stability of the Witt ring, (I 3 F)_red = 2(I 2 F)_red, we will prove stronger classification theorems by only requiring reduced stability on a certain subset of orderings of F, depending on the algebra and type of involution. Finally, we will prove applications of Parimala's exact sequence of Witt groups, including a Pfister local-global principle for hermitian forms.