# Linear representation theory of cyclic group:Z5

From Groupprops

## Contents |

This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z5.

View linear representation theory of particular groups | View other specific information about cyclic group:Z5

## Summary

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

degrees of irreducible representations over a splitting field | 1,1,1,1,1 maximum: 1, lcm: 1, number: 5, sum of squares: 5 |

Schur index values of irreducible representations | 1,1,1,1,1 |

condition for a field to be a splitting field | characteristic not equal to 5, must contain a primitive fifth root of unity, or equivalently, the polynomial must split. For a finite field of size , equivalent formulation: 5 must divide . |

smallest ring of realization (characteristic zero) | or , integral extension of the ring of integers of degree 4 |

smallest field of realization (characteristic zero) | or , cyclotomicextension of of degree 4 |

smallest size splitting field | field:F11, i.e., field of 11 elements |

degrees of irreducible representations over the field of real numbers , and more generally over a field where splits as a product of two irreducible quadratics | 1,2,2 maximum: 2, lcm: 2, number: 3 |

degrees of irreducible representations over the field of rational numbers , and more generally over a field where is irreducible | 1,4 |

## Family contexts

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

finite cyclic group | 5 | linear representation theory of finite cyclic groups |