Methods in Theoretical Physics problem (a) Find the five fifth roots of unity. (b) Sketch them in the complex plane. (c) What is the sum of the five roots? (d) What is the product of the five roots? Solution We want to solve the equation x 5 ?1=(x?1)(x 4 +x 3 +x 2 +x+1)=0 Then we are interested in solving x 4 +x 3 +x 2 +x+1=0. First, we divide everything by x 2 to get x 2 +x+1+1/x+1/x 2 =(x+1/x)+(x+1/x) 2 ?1 = u 2 +u?1=0 We noted (x+x ?1 ) 2 ?2=x 2 +x ?2 and substituted xu 2 +u=1+x ?1 =u The solutions for u 2 +u=1 are ?? and ??1, ? being the golden ratio. We now have two equations: x+1/x=?? ? x 2 +?x+1=0 ? x=[±sqrt(??3)??]/2 x+1/x=??1 ? x 2 +(1??)x+1=0? x=[±sqrt(???2)+??1]/2 .