[hidroto]

I am no mathematician, but i think you may be overstating Galois result. it says that you cant write a single closed form expression for the roots of any quintic using only (+,-,*,/,nth roots). This does not necessarily stop you from expressing each root individually with the standard algebraic operations.

[gizmo686]

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.

[hidroto]

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.

[mswtk]

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.