# Linear representation theory of cyclic group:Z8

## Contents

View linear representation theory of particular groups | View other specific information about cyclic group:Z8

## Summary

Item Value
degrees of irreducible representations over a splitting field 1,1,1,1,1,1,1,1 (1 occurs 8 times)
maximum: 1, lcm: 1, number: 8, sum of squares: 8
Schur index values of irreducible representations over a splitting field 1,1,1,1,1,1,1,1
Condition for a field to be a splitting field characteristic not 2, and $x^4 + 1$ should split completely.
For finite field of size $q$, equivalent to 8 dividing $q - 1$.
Smallest ring of realization (characteristic zero) $\mathbb{Z}[(1 + i)/\sqrt{2}]$, i.e., $\mathbb{Z}[e^{\pi i/4}]$ or $\mathbb{Z}[x]/(x^4 + 1)$
Smallest field of realization (characteristic zero) $\mathbb{Q}((1 + i)/\sqrt{2})$ or $\mathbb{Q}(i,\sqrt{2}))$ i.e., $\mathbb{Q}(e^{\pi i/4})$ or $\mathbb{Q}[x]/(x^4 + 1)$
Smallest size splitting field field:F9, i.e., the field of size nine
degrees of irreducible representations over reals 1,1,2,2,2
maximum: 2, lcm: 2, number: 5
degrees of irreducible representations over rationals 1,1,2,4
maximum: 2, lcm: 4, number: 4
orbit structure under automorphism group of representations over splitting field orbits of size 1,1,2,4
number: 4
orbit structure under action of Galois group for splitting field over rationals orbits of size 1,1,2,4
number: 4

## Family contexts

Family name Parameter values General discussion of linear representation theory of family
finite cyclic group 8 linear representation theory of finite cyclic groups