The quintic equation: Why is there no closed formula? We know that polynomials up to fourth degree have closed solutions using radicals. And we know that starting from the quintic no polynomial will have a closed solution using radicals.
Question 1: What I want to know is, why does this happen for the fifth order polynomial? What is so special about the number 5? I don't want a prove. I am looking for an intuitive explanation. 
Question 2: Is there a closed formula for higher order polynomials using other functions and operations? I heared that Felix Klein did something like that, but I could not find a closed formula for the solution.

If by ‘closed formula’, you mean a formula with radicals and ordinary arithmetic operations, the general answer comes from Galois theory:
One consider the Galois group of the equation, i.e. the group of automorphisms of the splitting  field of the polynomial. Suppose the polynomial has degree $n$. As an automorphism of this splitting field maps a root of the equation to another root, it is a subgroup $G$ of the symmetric  group $S_n$.
Now the equation is ‘solvable by radicals’ if and only if $G$ is a solvable group.
This means there is a sequence of subgroups:
$$\{e\}=G_0\subset G_1\subset\dots\subset G_r=G$$
such that each $G_i$ is normal in $G_{i+1}$ and $G_{i+1}/G_i$ is abelian.
It happens that for the general polynomial of degree $n$ (i.e.  with indeterminate coefficients) the Galois group is $S_n$, and that the alternating group $A_n$ is simple for $n\ge 5$, and of course, not abelian. Hence $S_n$ is not solvable.
