# Linear representation theory of cyclic group:Z4

From Groupprops

## Contents |

This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z4.

View linear representation theory of particular groups | View other specific information about cyclic group:Z4

## Summary

This article gives information on the linear representation theory of cyclic group:Z4, the cyclic group of order four.

Item | Value |
---|---|

degrees of irreducible representations over a splitting field | 1,1,1,1 maximum: 1, lcm: 1, number: 4, sum of squares: 4 |

Schur index values of irreducible representations over a splitting field | 1,1,1,1 |

smallest ring of realization (characteristic zero) | , same as , ring of Gaussian integers. Quadratic integral extension of |

smallest field of realization (characteristic zero) | , same as . Cyclotomic quadratic extension of . |

condition for a field to be a splitting field | Characteristic not two, and contains a square root of , i.e., a primitive fourth root of unity. Equivalently, the polynomial splits. |

degrees of irreducible representations over a field not a splitting field (includes case of , the field of real numbers, and , the field of rational numbers) | 1,1,2 |

smallest size splitting field | field:F5, i.e., the field with five elements. |

## Family contexts

Family name | Parameter values | General discussion of linear representation theory of family |
---|---|---|

finite cyclic group | 4 | linear representation theory of finite cyclic groups |