Let t be a complex number that is also a fifth root of unity. Prove the following:

Question

Let $t\in C$ be a fifth root of unity (that is, $t^5=1$). Prove that if $t\neq 1$ , then $t^4+t^3+t^2+t+1=0$.

Answer 1

Observe that $0=t^5-1=(t-1)(t^4+t^3+t^2+t+1)$. Since $t-1\neq 0$ we get the result.