# Multiplicative group of a field modulo squares

From Groupprops

## Definition

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.

## Particular cases

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 |

## Applications

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 .