Element of finite order is semisimple and eigenvalues are roots of unity

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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

Suppose k is a field, V is a finite-dimensional vector space over k and g is an element in GL(V) such that there exists n with g^n equal to the identity matrix. Then:

  • All the eigenvalues of g over the algebraic closure of k are n^{th} roots of unity.
  • Suppose that either k has characteristic zero or n is relatively prime to the characteristic of k. g is semisimple, i.e. it is diagonalizable over the algebraic closure of k.

Applications

Proof

g satisfies the polynomial x^n - 1, hence the minimal polynomial of g must divide this polynomial. So, every eigenvalue of g must satisfy the polynomial x^n - 1, hence must be a root of unity.

If n is relatively prime to the characteristic, then the polynomial x^n - 1 has no repeated roots, hence the minimal polynomial of g has no repeated roots. So, g is semisimple, i.e. it is diagonalizable over the algebraic closure.