Square class

In mathematics, specifically abstract algebra, a square class of a field formula_1 is an element of the square class group, the quotient group formula_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 "xy"2 where "x" is some particular fixed element and "y" ranges over all nonzero field elements.
For instance, if formula_3, the field of real numbers, then formula_4 is just the group of all nonzero real numbers (with the multiplication operation) and formula_5 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.
Square classes are frequently studied in relation to the theory of quadratic forms. The reason is that if formula_6 is an formula_1-vector space and formula_8 is a quadratic form and formula_9 is an element of formula_6 such that formula_11, then for all formula_12, formula_13 and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents.
Every element of the square class group is an involution. It follows that, if the number of square classes of a field is finite, it must be a power of two.