Algebra for Symbolic Computation pp 141-169 | Cite as

# The discrete Fourier transform

Chapter

## Abstract

The with In particular, for 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.

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

*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)

*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)

## Keywords

Abelian Group Cyclic Group Discrete Fourier Transform Group Algebra Orthogonality Relation
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## Copyright information

© Springer-Verlag Italia 2012