# The discrete Fourier transform

• Antonio Machì
Chapter
Part of the UNITEXT book series (UNITEXT)

## 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.

## Keywords

Abelian Group Cyclic Group Discrete Fourier Transform Group Algebra Orthogonality Relation
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