# Finite field

From Groupprops

## Definition

A **finite field** is a field with only finitely many elements.

Some key facts about finite fields include:

- The characteristic of a finite field must be a prime number.
- The size of a finite field must be a prime power. In fact, it must be a power of the prime number that is the characteristic of the field.
- For any prime power, there is a unique (up to isomorphism) finite field whose size equals that prime power.

Combining the above key facts, we denote, for any prime power , the unique finite field of size by the symbols or . (Note that GF stands for *Galois field* in recognition of Galois's pioneering work in field theory).

## Particular cases

Field size | Underlying prime (field characteristic) | , i.e, the number such that | Field |
---|---|---|---|

2 | 2 | 1 | field:F2 |

3 | 3 | 1 | field:F3 |

4 | 2 | 2 | field:F4 |

5 | 5 | 1 | field:F5 |

7 | 7 | 1 | field:F7 |

8 | 2 | 3 | field:F8 |

9 | 3 | 2 | field:F9 |

11 | 11 | 1 | field:F11 |

13 | 13 | 1 | field:F13 |

16 | 2 | 4 | field:F16 |

17 | 17 | 1 | field:F17 |