Compute $(1+\alpha^4)(1+\alpha^3)(1+\alpha^2)(1+\alpha)$ where $\alpha$ is the complex 5th root of unity with the smallest positive principal argument

Question

I just started on the topic Complex Numbers and there is a question that I am stuck on.

The question is:

If $\alpha$ is a complex 5th root of unity with the smallest positive principal argument, determine the value of $\mathbf(1+\alpha^4)(1+\alpha^3)(1+\alpha^2)(1+\alpha)$

From what I understand, I'm supposed to start with a^5=1 and that the smallest positive argument should be 2π/5 To be exact, I got the roots $\mathbf{e^{\frac{2ki\pi}{5}}}$ in which k is from 0 to 4. After that, I have no idea how to proceed.

Answer 1

Hint: the five $5^{th}$ roots of unity are $\,1,\alpha,\alpha^2,\alpha^3,\alpha^4\,$. Also, if $\,\beta\,$ is a root of $\,z^5-1\,$ then $\,1+\beta\,$ is a root of $\,(z-1)^5-1\,$, and therefore $\,(1+1)(1+\alpha)(1+\alpha^2)(1+\alpha^3)(1+\alpha^4)\,$ is the product of the five roots of $\,(z-1)^5-1\,$.

Answer 2

$$S=\mathbf(1+\alpha^4)(1+\alpha^3)(1+\alpha^2)(1+\alpha)$$ $$S(1-\alpha)=\mathbf(1+\alpha^4)(1+\alpha^3)(1+\alpha^2)(1+\alpha)(1-\alpha)$$ $$S(1-\alpha)=(1-\alpha^8)(1+\alpha^3)$$ $$S(1-\alpha)=(1-\alpha^3)(1+\alpha^3)$$ $$S(1-\alpha)=(1-\alpha^6)$$ $$S(1-\alpha)=(1-\alpha)$$ $$S=1$$