UC Santa Barbara

The Witt ring of a field gives the structure of the isometry classes of quadratic forms over that field. In particular, the Witt ring provides an algebraic invariant for fields away from characteristic $2$, which also allow us to study the orderings we can put on that field. During the latter quarter of the 20th century, abstract Witt rings, a wider class of rings that had the structure of a Witt ring but constructed independently from fields, were introduced. In this thesis, we will use what is known about the structure of Witt rings over quadratic extensions of fields in order to come up with an analog that extends over to abstract Witt rings.