Linear representation theory of cyclic group:Z4

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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 \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