Classification, construction, and similitudes of quadratic forms

We present a theory of classifying quadratic forms over an algebraic number field which is a reformulation of Eichler's methods and different from the traditional theory of Hasse. As applications we determine all possible values of the discriminant of an integer-valued quadratic form over Q with a given dimension and index, and also the number of genera of such forms. We also investigate purely algebraic problems on the Clifford groups and similitudes of quadratic forms.