# Splitting field

*This term associates to every group, a corresponding field property. In other words, given a field, every field either has the property with respect to that group or does not have the property with respect to that group*

## Contents

## Definition

### In terms of realization of irreducible representations

A **splitting field** for a finite group is a field satisfying **both** the following two conditions:

- Every linear representation of the group is completely reducible. This is equivalent to saying that the characteristic of the field does not divide the order of the group.
- Any linear representation of the group over any extension of the given field, is equivalent to a linear representation over the field itself

### In terms of the character ring

A **splitting field** for a finite group is a field satisfying **both** the following two conditions:

- Every linear representation of the group is completely reducible. This is equivalent to saying that the characteristic of the field does not divide the order of the group.
- The character ring of the group over the field is equal to the character ring of the group over any extension of the field. Here, the character ring of a group over a field is the ring of -linear combinations of characters of representations of the group realizable over that field.

Note that in some alternative definitions, only condition (2) is imposed for being a *splitting field*, thus also including the *modular* case where not all representations are completely reducible.

## Examples

Group | Order | Smallest splitting field in characteristic zero | Necessary and sufficient condition for splitting field in characteristic coprime to group order | Condition on for field of size , coprime to group order |
---|---|---|---|---|

trivial group | 1 | field of rational numbers | any field | any |

cyclic group:Z2 | 2 | field of rational numbers | any field | any |

cyclic group:Z3 | 3 | -- need to adjoin primitive cuberoot of unity | any field where splits | divides |

cyclic group:Z4 | 4 | -- need to adjoin squareroot of | any field where splits | divides |

Klein four-group | 4 | field of rational numbers | any field | any |

cyclic group:Z5 | 5 | any field where splits, or has a root | divides | |

symmetric group:S3 | 6 | any field | any | |

cyclic group:Z6 | 6 | -- need to adjoin primitive cuberoot of unity | any field where splits | divides |

group of prime order | any field where splits, equivalently, has a root | divides | ||

cyclic group of order | , where is the cyclotomic polynomial | any field where splits | divides | |

dihedral group:D8 | 8 | field of rational numbers | any field | any |

## Relation with other properties

### Stronger properties

- Sufficiently large field:
`For full proof, refer: Sufficiently large implies splitting` - Minimal splitting field: A splitting field not containing any smaller splitting field.