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**Extra resources for Fundamental problems in algorithmic algebra**

**Example text**

We use some other algebraic structures in addition to the ones introduced in Lecture 0, §1: GF (pm ) Zn Mm,n (R) Mn (R) = = = = Galois field of order pm , p prime, integers modulo n ≥ 1, m by n matrices over a ring R, Mn,n (R). Finite structures such as GF (pm ) and Zn have independent interest, but they also turn out to be important for algorithms in infinite structures such as Z. §1. The Discrete Fourier Transform The key to fast multiplication of integers and polynomials is the discrete Fourier transform.

Note that ω K = 2L ≡ −1(mod M ). , it is a (2K)th root of unity. To show that it is in fact a primitive root, we must show ω j ≡ 1 for j = 1, . . , (2K − 1). If j ≤ K then ω j = 2Lj/K ≤ 2L < M so clearly ω j ≡ 1. If j > K then ω j = −ω j−K where j − K ∈ {1, . . , K − 1}. Again, ω j−K < 2L ≡ −1 and so −ω j−K ≡ 1. D. We next need the equivalent of the cancellation property (Lemma 1). The original proof is invalid since ZM is not necessarily an integral domain (see remarks at the end of this section).

R. Algazi. A unified treatment of discrete fast unitary transforms. SIAM J. Computing, 6(4):700–717, 1977. [67] E. Frank. Continued fractions, lectures by Dr. E. Frank. Technical report, Numerical Analysis Research, University of California, Los Angeles, August 23, 1957. [68] J. Friedman. On the convergence of Newton’s method. Journal of Complexity, 5:12–33, 1989. [69] F. R. Gantmacher. The Theory of Matrices, volume 1. , New York, 1959. [70] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky.