Linear representation theory of cyclic group:Z8

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