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Basically I had a problem where I needed to find the sum of the squares of solutions of this $A = \{|z^n+1/z^n|: n\in \Bbb N ,\: z \in \Bbb C\,\text{ and }\, z^4+z^3+z^2+z^1+1 = 0 \} $.

How I understand these equations are done is amplifying with $(z-1)$ so you get $z^5-1=0$ but then it gets confusing for me We know $z=1$ can't be a solution but somehow the problem is divide in 5 cases, focusing on $z^{5k} = 1 $.

So for example $n=5k$, we have $|z^{5k+1}/z^{5k}|$ that will be $|1+1| = 2$ which is apparently a solution. And it keeps going for $n=5k+1$ until $n=5k+4$... I want to understand why we do this and how you solve for the other cases,from $n=5k+1$ to $n=5k+4$ I have a photo of the solution this way but I don't know how to post it here.

Bernard
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VlAd TbK
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1 Answers1

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Hint: $z^5-1 = (z-1)(z^4+z^3+z^2+z+1)$ and so the solutions of $z^4+z^3+z^2+z+1=0$ are the primitive 5th roots of unity.

Wuestenfux
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