Question

Prove the given definition, for parts a) through c).

Lemma 9.3.5 (Orthogonality Lemma). Fir N and let w-wN-e2mi/N be the natural primitive Nth root of unity in C. Fort Z/(N), we have: N-1 ktN ift-0 (mod N), 0 otherwise. Lukt (9.3.5) k-0 9.3.2. (Proves Lemma 9.3.5) Fix N є N, and let w-e2m/N. Let f(x)-r"-1. o510 (a) Explain why N-1 (9.3.9) (Suggestion: Try writing out the sum as 1 +z+....) (b) Explain why for any t є z/(N), fw)-0. (c) Prove that for any t є z/(N), 11 0, N-1 İft--0 (mod N), ift 0 (mod N). kt 9.3.10) 0 Suggestion: Use the fact that a, b e C, if ab 0, then either a 0 or b 0

Answer 1

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