This problem illustrates how to do the Fourier Transform (FT) in modular arithmetic, for example, modulo 7.
(a) There is a number a such that all the powers ω, ω2,...,ω6 are distinct (modulo 7). Find this ω, and show that ω + ω2 + + ω6 = 0. (Interestingly, for any prime modulus there is such a number.)
(b) Using the matrix form of the FT, produce the transform of the sequence (0,1,1,1, 5,2) modulo 7; that is, multiply this vector by the matrix M6(ω), for the value of ω you found earlier. In the matrix multiplication, all calculations should be performed modulo 7.
(c) Write down the matrix necessary to perform the inverse FT. Show that multiplying by this matrix returns the original sequence. (Again all arithmetic should be performed modulo 7.)
(d) Now show how to multiply the polynomials x2 + x + 1 and x3 + 2x − 1 using the FT modulo 7
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