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Why is it so hard to find the roots of polynomial equations?
For polynomials (with real coefficients), in degrees 2, 3, 4, there are the quadratic, cubic, and quartic formula, though the quartic formula is extremely long, so what makes degree 5 special that makes writing down a formula impossible?
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$\begingroup$There are some special types quintics which you can solve in radicals (5th and lower roots), but there are some other special ones whose roots have a strange type of symmetry that the expressions we form with radicals can't capture.
$\endgroup$ 1 $\begingroup$There certainly can be (and is) a formula. It's just not a formula with radicals. Others have mentioned why you can't do it with radicals, having to do with the solvable symmetries, but don't confuse this as to there being no formula at all. A good start might be to investigate Bring Radicals.
$\endgroup$ 1 $\begingroup$I don't think it's so much that $5$ is special but that $2,3$ and $4$ are exceptional. For an irreducible polynomial $p(x) \in \mathbb Q[x]$ there's a general expression for the roots of $p(x)$ in terms of radicals if and only if $\deg p(x) <5$. Now what exactly makes $2,3$ and $4$ different from every other numbers? Well we need some Galois theory and some group theory. To an irreducible polynomial you can associate a group $G$ called it's Galois Group. As it turns out a polynomial is solvable by radicals if and only if its Galois group is solvable. Now because of some elementary group actions arguments it turns out that a $G$ is a subgroup of $S_{\deg p(x)}$ that is the symmetric group on $\deg p(x)$ elements. So the exceptional part is that $S_n$ is only solvable when $n<5$.
That there are polynomials with Galois group $S_n$ is also a somewhat non-trivial fact. Essentially at low degrees nothing can go wrong with solvability, but at higher degrees everything falls apart.
$\endgroup$ 5 $\begingroup$Without getting into details, the reason five is special (i.e. that there is no formula for the roots in terms of the coefficients, the four arithmetic operations and radicals) is that the group $A_5$ of all even permutations of 5 letters is the smallest non-abelian simple group.
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