(a) Let n be odd and ω a primitive nth root of 1 (means that itsorder is n). Show this implies that −ω is a primitive 2nth root of1. Prove the converse: Let n be odd and ω a primitive 2nth root of1. Show −ω is a primitive nth root of 1. (b) Recall that the nthcyclotomic polynomial is deï¬ned as Φn(x) = Y gcd(k,n)=1 (x−ωk)where k ranges over 1,...,n−1 and ωk = e2Ï€ik/n is a primitive nthroot of 1. Compute Φ8(x) and Φ9(x), writing them out with Zcoefficients. Show your steps.
