Linear representation theory of cyclic group:Z5

This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z5.
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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 $\mathbb {R}$, 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