Find the value of this logarithmic expression involving fifth root of unity.

Question

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$

Thanks in anticipation for your help in solving this!

Answer 1

HINT:

$$1+\alpha+\alpha^2+\alpha^3=\frac{1-\alpha^4}{1-\alpha}=\frac{1-\frac1\alpha}{1-\alpha}\text{ as }\alpha^5=1$$

$$\implies 1+\alpha+\alpha^2+\alpha^3=\frac{1-\alpha}{-\alpha(1-\alpha)}=-\frac1\alpha\text{ as }\alpha\ne1$$