I'm doing exercises on Bonnafe's beautiful book Representations of $SL_2(\mathbb{F}_q)$. The first exercise is:
1.1. Let $k$ be the subfield of $\mathbb{F}_{q}$ generated by $\{ \operatorname{Tr}_{2}(\xi):\xi\in \mathbb{F}_{q^2} ~,\xi^{q+1}=1\}$. Show that $k=\mathbb{F}_{q}$. (Hint: Set $q^{\prime}=|k|$ and show that, if $\xi \in \mu_{q+1}:=\{a\in\mathbb{F}_{q^2} ~:a^{q+1}=1\}$, then $1+\xi^{2}+\xi^{ q^{\prime}}+\xi ^{q^{\prime}+1}=0$.)
I met some difficulty, as follows.
If $\xi\in\mathbb{F}_{q^2}$, then $\operatorname{Tr}(\xi):=\xi+\xi^q$. Then by the hint we have $$(\xi+\xi^q)^{q'}=\xi+\xi^q.$$ Expanding this, and remember that $\xi^q=\xi^{-1}$, we obtain $\xi+\xi^{-1}=\xi^{q'}+\xi^{-q'}$, and this is $1+\xi^{2}=\xi^{q'+1}+\xi^{-q'+1}$. Still I couldn't see how this is equivalent to the hint.
By the way, even assuming the claim of the hint. I still had no idea how to deduce the result of the exercise. Can anybody give me some help? Any suggestion or hint will be welcome. Thanks a lot in advance!