# Square class

In abstract algebra, a square class of a field (mathematics) $F$ is an element of the square class group, the quotient group $F^\times/ F^{\times 2}$ 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 $F=\mathbb{R}$, the field of real numbers, then $F^\times$ is just the group of all nonzero real numbers (with the multiplication operation) and $F^{\times 2}$ 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 $V$ is an $F$-vector space and $q:V \to F$ is a quadratic form and $v$ is an element of $V$ such that $q(v) = a \in F^\times$, then for all $u \in F^\times$, $q(uv) = au^2$ and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents.