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:
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:
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
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.
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- 1.
For simplicity, here and in what follows we denote by z the column vector formed by the elements of the n-tuple z, rather than by z t (t denoting transposition), as it should be.
- 2.
According to some authors, this U is the Fourier matrix; according to others the Fourier matrix is Ū,and (5.6)the Fourier transform,i:e the one given by F.
- 3.
By “operations” we shall mean multiplications.
- 4.
The technique of subdividing a problem in independent smaller problems is known as “divide et impera” (“divide and conquer”).
- 5.
Following the prevailing custom we denote here the principal character by χ 1 and not by χ 0; consequently, in this section we label the e i s starting with e 1, rather than with e 0 as we have previously done.
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© 2012 Springer-Verlag Italia
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Machì, A. (2012). The discrete Fourier transform. In: Algebra for Symbolic Computation. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2397-0_5
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DOI: https://doi.org/10.1007/978-88-470-2397-0_5
Publisher Name: Springer, Milano
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