# Linear representation theory of 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 $\R$, 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.

## Family contexts

Family name Parameter values General discussion of linear representation theory of family
finite cyclic group 4 linear representation theory of finite cyclic groups