Linear representation theory of cyclic group:Z8

This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z8.
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