The discrete Fourier transform

Abstract

The n-th roots of unity are the roots of the polynomial x ⁿ — 1 in the complex field. We know that they are all distinct because the polynomial is coprime with its derivative, and that they are all powers of one of them, a primitive root ω = e ^2πi/n:

$$ 1,w,{w}^2,\dots, {w}^{n-1}, $$

with w ⁿ = 1. (Recall that the primitive roots w ^k are those for which (k, n) = 1, so that their number is φ(η), where φ is Euler’s function.) Since 1 is a root of x ⁿ−1, this polynomial is divisible by x−1, with quotient 1+x+x ² +…+x ⁿ⁻¹, and therefore all the w ^k, with k ≠ 0 (or, actually, k ≠ 0 mod n), satisfy the equation:

$$ 1+x+{x}^2+\cdot \cdot \cdot +{x}^{n-1}=0. $$

(5.1)

In particular, for x = w we see that the sum of all the n-th roots of unity is zero, 1 + w + w ² +···+ w ^n-1 = 0. Moreover, since |w| = 1, we have |w|^k = 1, so that ${w}^k{\overline{w}}^k={\left|{w}^k\right|}^2=1$, from which

$$ {\overline{w}}^k=\frac{1}{w^k}={w}^{-k}. $$

(5.2)

We use formulas (5.1) and (5.2) to prove the following theorem; despite its simplicity, it will be of fundamental importance for the entire discussion.