Characters are cyclotomic integers

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This page describes a useful fact in character theory/linear representation theory arising from rudimentary linear algebra
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Statement

Let G be a finite group, and k be any field. Then, the character of any finite-dimensional linear representation of G over k takes, at every element of the group, a value that is a cyclotomic integer over the prime subfield of k: in other words, it is in the subring generated by all the roots of unity.

(The statement, does not require the characteristic of k to not divide the order of the group).

Related facts

Applications

  • Characters are algebraic integers: Any cyclotomic integer is an algebraic integer, so every character takes values in algebraic integers.
  • If, over a field of characteristic zero, a character takes a rational value, that value must be an integer.

Other related facts

Proof

Given: A finite group G, field k, a finite-dimensional linear representation (V,\rho) of G over k. \chi is the character of \rho.

To prove: For any g, \chi(g) is a cyclotomic integer over the prime subfield of k.

Proof: \chi(g) is the sum of eigenvalues of \rho(g), counted with multiplicity, in the algebraic closure of k. Since G is finite, g has finite order, so \rho(g) has finite order, and hence \rho(g) satisfies a polynomial of the form x^n - 1. Hence, every eigenvalue of \rho(g) is a n^{th} root of unity, so \chi(g) is a sum of n^{th} roots of unity. Hence, \chi(g) is a cyclotomic integer.