Linear representation theory of cyclic group:Z4

This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z4.
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Summary

This article gives information on the linear representation theory of cyclic group:Z4, the cyclic group of order four.

Item	Value
degrees of irreducible representations over a splitting field	1,1,1,1 maximum: 1, lcm: 1, number: 4, sum of squares: 4
Schur index values of irreducible representations over a splitting field	1,1,1,1
smallest ring of realization (characteristic zero)	$\mathbb {Z} [i]$, same as $\mathbb {Z} [x]/(x^{2}+1)$, ring of Gaussian integers. Quadratic integral extension of $\mathbb {Z}$
smallest field of realization (characteristic zero)	$\mathbb {Q} (i)$, same as $\mathbb {Q} [x]/(x^{2}+1)$. Cyclotomic quadratic extension of $\mathbb {Q}$.
condition for a field to be a splitting field	Characteristic not two, and contains a square root of $-1$, i.e., a primitive fourth root of unity. Equivalently, the polynomial $x^{2}+1$ splits.
degrees of irreducible representations over a field not a splitting field (includes case of $\mathbb {R}$, the field of real numbers, and $\mathbb {Q}$, the field of rational numbers)	1,1,2
smallest size splitting field	field:F5, i.e., the field with five elements.