# Square class

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{{Short description|In a commutative ring, an equivalence class modulo squares}}
In mathematics, specifically [abstract algebra](/source/abstract_algebra), a '''square class''' of a [field](/source/field_(mathematics)) <math>F</math> is an element of the '''square class group''', the [quotient group](/source/quotient_group) <math>F^\times/ F^{\times 2}</math> of the [multiplicative group](/source/multiplicative_group) of nonzero elements in the field modulo the [square](/source/Square_(algebra)) elements of the field. Each square class is a [subset](/source/subset) of the nonzero elements (a [coset](/source/coset) of the multiplicative group) consisting of the elements of the form ''xy''<sup>2</sup> where ''x'' is some particular fixed element and ''y'' ranges over all nonzero field elements.<ref name="salzmann">{{citation|title=The Classical Fields: Structural Features of the Real and Rational Numbers|volume=112|series=Encyclopedia of Mathematics and its Applications|first=H.|last=Salzmann|publisher=Cambridge University Press|year=2007|isbn=9780521865166|page=295|url=https://books.google.com/books?id=XQXiSHXkQDcC&pg=PA295}}.</ref>

For instance, if <math>F=\mathbb{R}</math>, the field of [real number](/source/real_number)s, then <math>F^\times</math> is just the group of all nonzero real numbers (with the multiplication operation) and <math>F^{\times 2}</math> is the [subgroup](/source/subgroup) of [positive number](/source/positive_number)s (as every positive number has a real [square root](/source/square_root)). The quotient of these two groups is a group with two elements, corresponding to two [coset](/source/coset)s: 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.<ref name="salzmann"/>

Square classes are frequently studied in relation to the theory of [quadratic form](/source/quadratic_form)s.<ref name="Szymiczek"/> The reason is that if <math>V</math> is an <math>F</math>-[vector space](/source/vector_space) and <math>q:V \to F</math> is a quadratic form and <math>v</math> is an element of <math>V</math> such that <math>q(v) = a \in F^\times</math>, then for all <math>u \in F^\times</math>, <math>q(uv) = au^2</math> 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](/source/Involution_(mathematics)). It follows that, if the number of square classes of a field is finite, it must be a [power of two](/source/power_of_two).<ref name="Szymiczek">{{citation|title=Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms|volume=7|series=Algebra, logic, and applications|first=Kazimierz|last=Szymiczek|publisher=CRC Press|year=1997|isbn=9789056990763|pages=29, 109|url=https://books.google.com/books?id=CcM8_iiGPxAC&pg=PA29}}.</ref>

==References==
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Category:Field theory

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