{{refimprove|date=August 2016}}
{{refimprove|date=August 2016}}

In [mathematics](/wiki/Mathematics "Mathematics"), a **principal *n*\-th root of unity** (where *n* is a positive [integer](/wiki/Integer "Integer")) of a [ring](/wiki/Ring_%28mathematics%29 "Ring (mathematics)") is an element \\alpha satisfying the equations

 \\begin{align}
\& \\alpha^n \= 1 \\\\
\& \\sum\_{j\=0}^{n\-1} \\alpha^{jk} \= 0 \\text{ for } 1 \\leq k \< n
\\end{align}
In an [integral domain](/wiki/Integral_domain "Integral domain"), every [primitive](/wiki/Primitive_root_of_unity "Primitive root of unity") *n*\-th [root of unity](/wiki/Root_of_unity "Root of unity") is also a principal n\-th root of unity. In any ring, if *n* is a [power of 2](/wiki/Power_of_2 "Power of 2"), then any *n*/2\-th root of −1 is a principal *n*\-th root of unity.

A non\-example is 3 in the [ring of integers modulo 26](/wiki/Ring_of_integers_modulo_n "Ring of integers modulo n"); while 3^3 \\equiv 1 \\pmod{26} and thus 3 is a [cube root of unity](/wiki/Cube_root_of_unity "Cube root of unity"), 1 \+ 3 \+ 3^2 \\equiv 13 \\pmod{26} meaning that it is not a principal cube root of unity.

The significance of a root of unity being *principal* is that it is a necessary condition for the theory of the [discrete Fourier transform](/wiki/Discrete_Fourier_transform_%28general%29 "Discrete Fourier transform (general)") to work out correctly.


How is a principal n-th root of unity defined in mathematics?