This dissertation is on sums-of-squares formulas, focusing on whether existence of a

formula depends on the base field and on methods for finding sums-of-squares formulas.

We introduce a new approach to the study of sums-of-squares formulas, showing that a

formula can be regarded as a solution to a system of polynomial equations and thus we can introduce the variety of sums-of-squares formula. This new perspective allows us to utilize tools of algebraic geometry in the study of sums-of-squares formulas, as well as raising new questions about the properties of this variety.

We use number theory and computational algebraic geometry to consider the question

of whether existence of sums-of-squares formulas depends on the base field. We are able to show independence in some cases, however, the general case remains an open question. Furthermore, we show that existence of a formula over an algebraically closed field is computable.

We introduce an algebraic group action on the variety of sums-of-squares formulas, which gives us another way to study the structure of this variety and raises new questions about the structure of the action.

Finally, we provide algorithms for finding sums-of-squares formulas over the integers and finite fields. These algorithms use previous results about formulas over the integers, as well as the group action on the scheme of sums-of-squares formulas.