Linear representation theory of cyclic group:Z5

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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 \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