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

(The statement, does not require the characteristic of 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 , field , a finite-dimensional linear representation of over . is the character of .

To prove: For any , is a cyclotomic integer over the prime subfield of .

Proof: is the sum of eigenvalues of , counted with multiplicity, in the algebraic closure of . Since is finite, has finite order, so has finite order, and hence satisfies a polynomial of the form . Hence, every eigenvalue of is a root of unity, so is a sum of roots of unity. Hence, is a cyclotomic integer.