Kaplansky's theorem on quadratic forms

Kaplansky's theorem on quadratic forms is a significant result in the theory of quadratic forms over rings, particularly concerning the values that can be obtained by quadratic forms over certain fields. The theorem specifically states conditions under which a quadratic form can be represented as the sum of squares of linear forms. In particular, one of the most notable facets of Kaplansky's work on quadratic forms relates to the representation of forms over the integers and over various fields.