Comment by gizmo686
2 months ago
I think you are thinking of the Abel–Ruffini impossibility theorum, which states that there is no general solution to polynomials of degree 5 or greater using only standard operations and radicals.
Galois went a step further and proved that there existed polynomials whose specific roots could not be so expressed. His proof also provided a relatively straightforward way to determine if a given polynomial qualified.
Thanks for the correction. It seems that all the layman’s explanations on Galois theory i have seen have been simplified to the point of being technically wrong, as well as underselling it.
Technically, the actual statement in Galois theory is even more general. Roughly, it says that, for a given polynomial over a field, if there exists an algorithm that computes the roots of this polynomial, using only addition, subtraction, multiplication, division and radicals, then a particular algebraic structure associated with this polynomial, called its Galois group, has to have a very regular structure.
So it's a bit stronger than the term "closed formula" implies. You can then show explicit examples of degree 5 polynomials which don't fulfill this condition, prove a quantitative statement that "almost all" degree 5 polynomials are like this, explain the difference between degree 4 and 5 in terms of group theory, etc.