# Field:F5

From Groupprops

## Definition

This is the unique field (up to isomorphism) having five elements. It is a prime field, and is the quotient of the ring of integers by the ideal of multiples of .

## Related groups

Group functor | Value | GAP ID |
---|---|---|

additive group | cyclic group:Z5 | (5,1) |

multiplicative group | cyclic group:Z4 | (4,1) |

general affine group of degree one | general affine group:GA(1,5) | (20,3) |

general linear group of degree two | general linear group:GL(2,5) | (480,218) |

special linear group of degree two | special linear group:SL(2,5) | (120,5) |

projective general linear group of degree two | symmetric group:S5 | (120,34) |

projective special linear group of degree two | alternating group:A5 | (60,5) |

## GAP implementation

The field can be defined using GAP's GF function:

`GF(5)`

It can also be defined using the ZmodnZ function:

`ZmodnZ(5)`