This talk took us through one of the most counterintuitive ideas: that not every polynomial can be factorised in terms of its solutions and that the barrier for that is so low as to be the cubic.
Starting with the quadratic, a base of knowledge we all know, Toller's exceptional talk then brought us to the cubic, explaining its quirks and how to reduce it to a form we all know, with a neat trick to turn it into a depressed cubic. He then used a clever method to split x into u+v and then made everything a quadratic.
After following through the algebra, we transitioned into the quartic, having solved a few examples with both real and complex roots. For the quartic, the method was more complicated but doable all the same. Yet he posed the idea that the quintic cannot be solved with a formula.
This took us into a tour of group theory, starting with what a group is and its symmetries, then moving to the symmetric group of n things. Then he posed a clever game where, knowing the roots of a shared equation, he would need to find out which roots we've swapped or whether we've swapped any at all. All the while, the slow, gradual increase in difficulty allowed for the talk to be easily understood. Having looked at the subgroups of the symmetric group, we then found out about regular subgroups, seeing a parallel between the methods used to solve our cubics and quartics and the methods used to split a symmetric group down into parts.
Finally, he left us with the idea that while S4 and S3 could be split up, S5 could not, leading us to the fact that the quintic cannot be solved with an equation, a powerful note to end on.
