On the notion of $cq$-genus of quadratic forms

The concept of $cq$-genus of quadratic forms is introduced, intermediate between the concepts of genus and spinor genus. This gives the possibility to identify properties of quadratic forms, dependent first of all on the structure of the group of the classes of the ground field. The fundamental properties of $cq$-genera, analogous in many respects with the properties of spinor genera, are considered. Applications of this concept to various aspects of the problem of the classification of quadratic forms are given.