Multiplicative group of a field modulo squares
Suppose is a field. The multiplicative group modulo squares of , denoted , is defined as the quotient group of the multiplicative group (the group of nonzero elements of under multiplication) by the subgroup .
The group is an elementary abelian 2-group, i.e., it is an abelian group in which every non-identity element has order two.
|Case for field||Conclusion for||Explicit description of cosets of in||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 in : 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,||Klein four-group||a basis is given by a quadratic nonresidue modulo , and the number itself||2|
|field of p-adic numbers,||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|
The group appears in a number of contexts. Some of these are given below:
- Element structure of special linear group of degree two over a field: The number of pieces into which the unipotent Jordan block conjugacy class splits in equal the order of the group . The group also determines, more indirectly, the splitting of some other conjugacy classes.
- Spinor norm is a homomorphism from an orthogonal group over to .