Principal root of unity
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In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element satisfying the equations
In an integral domain, every primitive n-th root of unity is also a principal -th root of unity. In any ring, if is a power of , then any -th root of is a principal -th root of unity.
A non-example is in the ring of integers modulo ; while and thus is a cube root of unity, 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 to work out correctly.
References[]
- Bini, D.; Pan, V. (1994), Polynomial and Matrix Computations, vol. 1, Boston, MA: Birkhäuser, p. 11
Categories:
- Algebraic numbers
- Cyclotomic fields
- Polynomials
- 1 (number)
- Complex numbers