> In 1963, the Russian mathematician Vladimir Arnold gave an alternative
topological proof of the unsolvability of the quintic in a series of lectures to high school kids in Moscow.<p>...<p>> Arnold’s insight was to show that if there is a radical formula for the roots of a general polynomial, then the “dance of the roots” cannot be overly complex, in the sense that the image of the monodromy map must be a solvable subgroup of . But, for example, the 1-parameter Brioschifamily of quintics
() = 5 + 103 + 452 + 2, ∈ ℂ
has monodromy group 5 (see Figure 1), which is certainly not solvable since it is simple, as we will see by relating it to the icosahedron in the next section. Hence the unsolvability of the quintic.<p>...of course, well within the reach of most high school students.