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.

Particular cases

Case for field $K$	Conclusion for $(K^{*})/(K^{*})^{2}$	Explicit description of cosets of $(K^{*})^{2}$ in $K^{*}$	Rank as a vector space over field:F2 (i.e., minimum size of generating set)
finite field of characteristic two	trivial group	every element is a square, so there is only one coset	0
finite field of characteristic not two	cyclic group:Z2	the two elements of this gorup correspond to the two cosets of $(K^{*})^{2}$ in $K^{*}$: the identity element corresponds to quadratic residues and the non-identity element corresponds to quadratic nonresidues	1
field of rational numbers	countable-dimensional elementary abelian group	the group has a basis comprising -1 and all the positive integer primes	countably infinite
field of real numbers	cyclic group:Z2	the two cosets are as follows: the coset for the identity element is the set of positive reals, and the coset for the non-identity element is the set of negative reals	1
field of p-adic numbers, $p\neq 2$	Klein four-group	a basis is given by a quadratic nonresidue modulo $p$, and the number $p$ itself	2
field of p-adic numbers, $p=2$	elementary abelian group:E8	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	3
algebraically closed field (any characteristic)	trivial group	every element is a square, so there is only one coset	0

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 $(K^{*})/(K^{*})^{2}$.