Relatedly, to this day I still don't know how distinguish a left-handed coordinate system from the right-handed one purely algebraically. Is the basis [(1,0,0), (0,1,0), (0,0,1)] left- or right-handed? I don't know without a picture! Does anyone?
"Left" and "right" are a dipole. Neither one can exist without the other, and they are symmetric. It's the same issue as we have with the conjugates discussed in Galois Theory.
In fact, in an algebraic (non-ordered/arithmetic/analytic) perspective, it's misleading to use the symbols + and - to label the conjugates in field extensions like sqrt2 and i. Left and Right are better names than + and - for those conjugate pairs. Only when we impose an arithmetic ordering (which is not needed in the theory of algebraic equalities) is it meaningful to use + and -: -sqrt(x) < 0 < +sqrt(x), where x is a positive real number.
(and when x is a negative real number, we immediately see the problem with - again: -i and i are not separable via ordering with respect to 0.)
“Let a_1, …, a_5 be the five roots of p(x).”
The complex numbers are essentially the theory of 2D-space. You are asking about 3D-space. The statement that you quoted tells you that you cannot distinguish between up and down in theory.
Now, 2D-space is part of every 3D-space. There are multiple ways to see this. The easiest is to just drop the z-coordinate.
Suppose you could distinguish left-handed and right-handed in 3D-space. In this case, you would have a way to distinguish up and down in the embedded 2D-space. However, you cannot do this distinction in 2D-space and therefore you cannot distinguish left-handed and right-handed in 3D-space.