Consider the map $F:\mathbb{C}^3\to\mathbb{C}^3$ given by $F(x,y,z)=(x+y+z,xy+yz+zx,xyz)$. Is the map $F$ surjective? Also, is it both open and closed?
The map seems like mapping any given triple of complex numbers to elementary symmetric polynomials in three varibales. Since the space of symmetric polynomials is the projective space, and in addition the three components of the image of $F$ are independent of each other, I think the space spanned by the range is the whole of $\mathbb{C}^3$. As regards the map being both open and closed, I think we must take an open ball in $\mathbb{C}^3$ and prove that its image is open; and take any convergent sequence in $\mathbb{C}^3$ and prove the corresponding image sequence converges. But how do we do those? Any hints? Thanks beforehand.
Hint: Choose any complex $a$, $b$ and $c$ and define $f(w) = w^3 - aw^2 + bw - c$. If $x$, $y$ and $z$ are its complex roots, consider how $a$, $b$ and $c$ relate to these values (i.e., use Vieta's formulas).