Galois

I came across a pretty cool application of group theory the other day and a pretty cool story to go along with it, so I thought I would share it with you. Évariste Galois was a French mathematician in the early 1800's. He is accredited with starting the fields of Galois theory and Galois connections. He is also thought to be the first person to use the word "group" as a mathematical term. Above all that, though, he was a very stupid, and stubborn man.

For almost as long as there has been mathematics, mathematicians have been concerned with the solutions to polynomials. So we studied for a while and came up with a general equation for second degree polynomials which was nice and pretty and everything. Of course, there was no stopping there, so we worked a little harder and came up a similar equation for third degree polynomials and fourth degree polynomials. If you click on the links you'll see that those equations are gigantic and impractical, but nevertheless, there they are. Then mathematicians started working on an equations for fifth degree polynomials but found it to be a lot harder. So they worked and worked and came up with a whole entire branch of mathematics called algebraic geometry to try to find this equation, but they couldn't do it. I guess the mathematicians at that time just sort of assumed that they weren't smart enough yet to figure it out, so they put it off for a while. Then, along came this Galois fella.

Galois was a very excellent mathematician and quite the playboy of his day. He lived during a time of political turmoil in France and he was pretty outspoken about the whole thing, so between that and being a young, hotshot mathematician he was a pretty popular dude. Unfortunately, he was also kind of a jerk and one day he was challenged to a duel at the age of 20. (I don't know by whom and I don't know why, but its not really important.) Well, Galois apparently wasn't that good with a gun and he knew it. Because he was so convinced of his impending death, that night he wrote a letter to a man named Auguste Chevalier detailing all of his work in mathematics. Well, as it happens, staying up all night and writing down a lifetime's worth of mathematics doesn't really help you in a duel, so he died the next day. Chevalier, for whatever reason, never read the letter. He never threw it away either, though, and twenty-four years after receiving it he died and the letter somehow found its way into the hands of another very brilliant mathematician, Joseph Liouville. Liouville did read it and inside was an entirely new approach to the solutions of polynomials which we now call Galois theory. Once Liouville filled in a few of the gaps, Galois' work had produced a brilliant and concise proof showing that there is no general solution to a fifth degree polynomial, which is very cool.

What Galois did was attach a group to a polynomial in a very natural way (that is, natural if you understand the basics of both group theory and algabraic topology) and produced an extremely cool theorem that not only explains why there is no general solution to a fifth degree polynomial, but also explains why there is such a solution for second, third, and fourth degree polynomials.

This is what I love so much about mathematics - not the very stupid people that it seems to attract, but the ingenious breakthroughs that tie seemingly unrelated things together into something beautiful.