# Linear representation theory of cyclic group:Z5

## Contents

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 $x^4 + x^3 + x^2 + x + 1$ must split.
For a finite field of size $q$, equivalent formulation: 5 must divide $q - 1$.
smallest ring of realization (characteristic zero) $\mathbb{Z}[e^{2\pi i/5}]$ or $\mathbb{Z}[x]/(x^4 + x^3 + x^2 + x + 1)$, integral extension of the ring of integers of degree 4
smallest field of realization (characteristic zero) $\mathbb{Q}(e^{2\pi i/5})$ or $\mathbb{Q}[x]/(x^4 + x^3 + x^2 + x + 1)$, cyclotomicextension of $\mathbb{Q}$ 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 $\R$, and more generally over a field where $x^4 + x^3 + x^2 + x + 1$ 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 $\mathbb{Q}$, and more generally over a field where $x^4 + x^3 + x^2 + x + 1$ 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