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.
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.