When you try to solve a degree n equation, there are n roots you have to find (in principle) and none of them is favoured over any of the others, which (in some metaphorical sense) means that you have to break an n-fold symmetry in order to write down the roots.

Now the symmetry group of the n roots becomes more and more complicated the larger n is. For n = 2 it is abelian (and very small!); for n = 3 and 4 it is still solvable (in the technical sense of group theory), which explains the existence of an explicit formula involving radicals (this is due to Galois, and is a part of so-called Galois theory); for n = 5 or more this group is non-solvable (in the technical sense of group theory), and this corresponds to the fact that there is no explicit formula involving radicals.

Summary: The complexity of the symmetry group of the n roots leads to a corresponding complexity in explicitly solving the equation.

When you try to solve a degree $n$ equation, there are $n$ roots you have to find (in principle) and none of them is favoured over any of the others, which (in some metaphorical sense) means that you have to break an $n$-fold symmetry in order to write down the roots.

Now the symmetry group of the n roots becomes more and more complicated the larger $n$ is. For $n = 2$, it is abelian (and very small!); for $n = 3$ and $4$ it is still solvable (in the technical sense of group theory), which explains the existence of an explicit formula involving radicals (this is due to Galois, and is a part of so-called Galois theory); for $n = 5$ or more this group is non-solvable (in the technical sense of group theory), and this corresponds to the fact that there is no explicit formula involving radicals.

Summary: The complexity of the symmetry group of the $n$ roots leads to a corresponding complexity in explicitly solving the equation.

When you try to solve a degree n equation, there are n roots you have to find (in principal) and none of them is favoured over any of the others, which (in some metaphorical sense) means that you have to break an n-fold symmetry in order to write down the roots.

Now the symmetry group of the n roots becomes more and more complicated the larger n is. For n = 2 it is abelian (and very small!); for n = 3 and 4 it is still solvable (in the technical sense of group theory), which explains the existence of an explicit formula involving radicals (this is due to Galois, and is a part of so-called Galois theory); for n = 5 or more this group is non-solvable (in the technical sense of group theory), and this corresponds to the fact that there is no explicit formula involving radicals.

Summary: The complexity of the symmetry group of the n roots leads to a corresponding complexity in explicitly solving the equation.

When you try to solve a degree n equation, there are n roots you have to find (in principle) and none of them is favoured over any of the others, which (in some metaphorical sense) means that you have to break an n-fold symmetry in order to write down the roots.

Now the symmetry group of the n roots becomes more and more complicated the larger n is. For n = 2 it is abelian (and very small!); for n = 3 and 4 it is still solvable (in the technical sense of group theory), which explains the existence of an explicit formula involving radicals (this is due to Galois, and is a part of so-called Galois theory); for n = 5 or more this group is non-solvable (in the technical sense of group theory), and this corresponds to the fact that there is no explicit formula involving radicals.

Summary: The complexity of the symmetry group of the n roots leads to a corresponding complexity in explicitly solving the equation.