solving polynomials is really hard…

$\mathbb{C}$ is algeabraically closed - What happens on the complex plane stay on the complex plane

There is no function that take coefficient and gives solution and is continuous.

And so can not be written with basic algrebra (indeed quadratic formula use $\pm$)

Square root can be extend to riemann surface for continuity (that returns 2 value opposite to each other).

Commutators

• 2 roots - uneeded
• 3 roots - 1 commutators
• 4 roots - more
• 5 roots (quintics) - 2x 3cycles - that connect in center - there is no finite number of nested roots (sqr) and cannot be expressed by commutators.
  • There is no algrebric formulla that solve quintics