# Square class

In abstract algebra, a square class of a field (mathematics) ${\displaystyle F}$ is an element of the square class group, the quotient group ${\displaystyle F^{\times }/F^{\times 2}}$ of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements.[1]
For instance, if ${\displaystyle F=\mathbb {R} }$, the field of real numbers, then ${\displaystyle F^{\times }}$ is just the group of all nonzero real numbers (with the multiplication operation) and ${\displaystyle F^{\times 2}}$ is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers.[1]
Square classes are frequently studied in relation to the theory of quadratic forms.[2] The reason is that if ${\displaystyle V}$ is an ${\displaystyle F}$-vector space and ${\displaystyle q:V\to F}$ is a quadratic form and ${\displaystyle v}$ is an element of ${\displaystyle V}$ such that ${\displaystyle q(v)=a\in F^{\times }}$, then for all ${\displaystyle u\in F^{\times }}$, ${\displaystyle q(uv)=au^{2}}$ and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents.