# 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, $p \ne 2$ 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] 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 $(K^*)/(K^*)^2$.