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where in $\mathbb{C}^2$ is $z=2+i$
I’m currently reading Rosenlicht’s book about manifolds.
I was solving exercise 3.11 from page 209, he states that the morphism $\varphi:(x,y)\mapsto(x,y,x^2-y^2)$ is surjective since
$$\varphi\left((2,2)+(1,2i)\right)=(2,2,8)=\varphi(1,1+i)=\varphi((1,0)+i)=\varphi((0,1))\in \varphi\left(\mathbb{C}^2-(\{0\}\times\mathbb{C})\right).$$
Ofcourse I know that $\mathbb{C}^2=\{(x,y)\in \mathbb{R}^2 : x^2+y^2=1\}$.
But why is $z=2+i$ a solution for that?
It’s at least true for $z=1+i$ or $z=3+2i$ but I want to find out if he made an error or if this is a necessary condition.
A:
If $w=2+i$, then
$$
z^2=(w+\overline{w})^2=(w^2+\overline{w}^2+2\cd
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