Root of unity

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a root of unity, occasionally called a <a href="/facts/Abraham_de_Moivre/m9hpa6Mb">de Moivre</a> number, is any <a href="/facts/Complex_number/2w7mqI7e">complex number</a> that yields 1 when <a href="/facts/Exponentiation/pac6QY2g">raised</a> to some positive <a href="/facts/Integer/PUwwrawx">integer</a> power n. Roots of unity are used in many branches of mathematics, and are especially important in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the theory of <a href="/facts/Group_character/WMp5Q6YE">group characters</a>, and the <a href="/facts/Discrete_Fourier_transform/uK9Tjsbl">discrete Fourier transform</a>.
</p><p>Roots of unity can be defined in any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. If the <a href="/facts/Characteristic_of_a_field/D2VzcQaG">characteristic</a> of the field is zero, the roots are complex numbers that are also <a href="/facts/Algebraic_integer/Qb0fD3R3">algebraic integers</a>. For fields with a positive characteristic, the roots belong to a <a href="/facts/Finite_field/UkMGJo3m">finite field</a>, and, <a href="/facts/Converse_(logic)/VhlUrxxx">conversely</a>, every nonzero element of a finite field is a root of unity. Any <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a> contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.
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Root of unity open-in-new

Root of unity