Suppose that instead of performing an n-element FFT over the field of complex numbers (where n is even), we use the ring ℤ_m of integers modulo m, where m = 2^tn/2 + 1 and t is an arbitrary positive integer. Use ω = 2^t instead of ω_n as a principal nth root of unity, modulo m. Prove that the DFT and the inverse DFT are well defined in this system.