Chapter 6: The Frustrated Doctor and the Sickly Genius

Many Mathematician had been searching for a solution to a quintic equation. One man named Paolo Ruffini developed a 500 page proof that solving any equation with a dregree higher than 4 was impossible. Mathematicians found this "daunting"; to devote so many hours into proving that something does NOT work. Rumours went around that there were mistakes in his proof, but no one would identify them so he could not defend himself. He knew that his proof was extremely complicated, so he tried again, which did not result any better.

"One reason Ruffini's work was not well received may have been it's novelty.
Like Lagrange, he based his investigations on the concept of a "permutation.""

A permutation is a way to rearrange some ordered list. An example of this would be the results of shuffling a pack of cards. The idea is that you'll end up with a totally random and unpredictable order of the cards. Because the number of permutations possible in a deck of cards, it makes it basically impossible to predict the outcome.



“Permutations arise in the theory of equations because the roots of a given
polynomial can be considered a list. Some very basic features of equations are
directly related to the effect of shuffling that list. The intuition is that the
equation "does not know" the order in which you listed its roots, so permuting
the roots should not make any important difference. In particular, the
coefficients of the equation should be fully symmetric expressions in the
roots-expressions that do not change when the roots are permuted."

This is basically explaining that in a given polynomial, the roots of the equation can be considered a list, so we can incorporate the idea of permutations when trying to solve. "Shuffling" that list is proven to have effects on the equation.


Ruffini also used another of Lagranges ideas. By multipling two permutations together, you would result in getting another permutation. For example, take the symbols a, b, and c. This has 6 permutations: abc, acb, bac, bca, cab, and cba. Now take one of these permutations, cba for example, and we can see this as just an ordered pair from the list. However, it can also be seen as a rule for rearranging the original set. This rule would be, reverse the order. We can apply this to any of the permutions; if we apply it to bca, we'll get acb. Therefore it would make sense that cba times bca would equal abc.



We can also combine the two into one by stacking them on top of eachother. There are two ways to do it. Also, the result depends on which permutation comes first.