Editing Finite field (section)

=== Roots of unity ===

Every nonzero element of a finite field is a [[root of unity]], as <math>x^{q-1}=1</math> for every nonzero element of <math>\mathrm{GF}(q)</math>.

If <math>n</math> is a positive integer, an <math>n</math>th '''primitive root of unity''' is a solution of the equation <math>x^n=1</math> that is not a solution of the equation <math>x^m=1</math> for any positive integer <math>m<n</math>. If <math>a</math> is a <math>n</math>th primitive root of unity in a field <math>F</math>, then <math>F</math> contains all the <math>n</math> roots of unity, which are <math>1,a,a^2,\ldots,a^{n-1}</math>.

The field <math>\mathrm{GF}(q)</math> contains a <math>n</math>th primitive root of unity if and only if <math>n</math> is a divisor of <math>q-1</math>; if <math>n</math> is a divisor of <math>q-1</math>, then the number of primitive <math>n</math>th roots of unity in <math>\mathrm{GF}(q)</math> is <math>\phi(n)</math> ([[Euler's totient function]]). The number of <math>n</math>th roots of unity in <math>\mathrm{GF}(q)</math> is <math>\mathrm{gcd}(n,q-1)</math>.

In a field of characteristic <math>p</math>, every <math>np</math>th root of unity is also a <math>n</math>th root of unity. It follows that primitive <math>np</math>th roots of unity never exist in a field of characteristic <math>p</math>.

On the other hand, if <math>n</math> is [[coprime]] to <math>p</math>, the roots of the <math>n</math>th [[cyclotomic polynomial]] are distinct in every field of characteristic <math>p</math>, as this polynomial is a divisor of <math>X^n-1</math>, whose [[discriminant]] <math>n^n</math> is nonzero modulo <math>p</math>. It follows that the <math>n</math>th [[cyclotomic polynomial]] factors over <math>\mathrm{GF}(q)</math> into distinct irreducible polynomials that have all the same degree, say <math>d</math>, and that <math>\mathrm{GF}(p^d)</math> is the smallest field of characteristic <math>p</math> that contains the <math>n</math>th primitive roots of unity.

When computing [[Modular representation theory|Brauer characters]], one uses the map <math>\alpha^k \mapsto \exp(2\pi i k / (q-1))</math> to map eigenvalues of a representation matrix to the complex numbers. Under this mapping, the base subfield <math>\mathrm{GF}(p)</math> consists of evenly spaced points around the unit circle (omitting zero).

[[File:Finite field with 25 elements and base subfield of five elements.jpg|thumb|300px|Finite field F_25 under map to complex roots of unity. Base subfield F_5 in red.]]