Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$.

Question

I'm starting to see complex numbers in algebra. I've missed a few classes and I have exercises similar to this one:

Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$.

It should break down at one moment to two cases, when $w=1$ and $w \not = 1$.

I'm not sure how to get there. So far I have that the expression above should be equal to $w+w^2+w^3+w^4+w^5+w^6$ if I didn't mess up along the way.

Could you give me any hint of what I should do next? (It's not for grading so a complete answer would be useful as well) Many thanks!

Answer 1

Your expression should evaluate to $w+w^2+w^3+w^4+w^5+w^6$.

For the case $w\neq1$, you can use that $1+w+w^2+w^3+w^4+w^5+w^6=0$$.

See for example: Intuitive understanding of why the sum of nth roots of unity is 0?

For the case $w=1$, simply evaluate $w+w^2+w^3+w^4+w^5+w^6=1+1^2+\ldots=?$.