Multiplicative group of a field modulo squares

Definition
Suppose $$K$$ is a field. The multiplicative group modulo squares of $$K$$, denoted $$K^*/(K^*)^2$$, is defined as the quotient group of the multiplicative group $$K^*$$ (the group of nonzero elements of $$K$$ under multiplication) by the subgroup $$(K^*)^2 = \{ x^2 \mid x \in K^* \}$$.

The group is an elementary abelian 2-group, i.e., it is an abelian group in which every non-identity element has order two.

Applications
The group $$K^*/(K^*)^2$$ appears in a number of contexts. Some of these are given below:


 * Element structure of special linear group of degree two over a field: The number of pieces into which the unipotent Jordan block conjugacy class splits in $$SL(2,K)$$ equal the order of the group $$K^*/(K^*)^2$$. The group also determines, more indirectly, the splitting of some other conjugacy classes.
 * Spinor norm is a homomorphism from an orthogonal group over $$K$$ to $$(K^*)/(K^*)^2$$.