Multiplicative group of a field modulo squares

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Definition

Suppose K is a field. The multiplicative group modulo squares of K, denoted K^*/(K^*)^2, is defined as the quotient group of the multiplicative group K^* (the group of nonzero elements of K under multiplication) by the subgroup (K^*)^2 = \{ x^2 \mid x \in K^* \}.

The group is an elementary abelian 2-group, i.e., it is an abelian group in which every non-identity element has order two.

Particular cases

Case for field K Conclusion for (K^*)/(K^*)^2 Explicit description of cosets of (K^*)^2 in K^* Rank as a vector space over field:F2 (i.e., minimum size of generating set)
finite field of characteristic two trivial group every element is a square, so there is only one coset 0
finite field of characteristic not two cyclic group:Z2 the two elements of this gorup correspond to the two cosets of (K^*)^2 in K^*: the identity element corresponds to quadratic residues and the non-identity element corresponds to quadratic nonresidues 1
field of rational numbers countable-dimensional elementary abelian group the group has a basis comprising -1 and all the positive integer primes countably infinite
field of real numbers cyclic group:Z2 the two cosets are as follows: the coset for the identity element is the set of positive reals, and the coset for the non-identity element is the set of negative reals 1
field of p-adic numbers, p \ne 2 Klein four-group a basis is given by a quadratic nonresidue modulo p, and the number p itself 2
field of p-adic numbers, p = 2 elementary abelian group:E8 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 3
algebraically closed field (any characteristic) trivial group every element is a square, so there is only one coset 0

Applications

The group K^*/(K^*)^2 appears in a number of contexts. Some of these are given below: