Couple of years ago when I first introduced my high school friends to abstract algebra, and the most glorious name associated with it, Evariste Galois. They weren’t that excited. In fact one chubby friend of mine called him the Gaylois (pronounced gay-loy-s) dude. Only I know how great this man is. So for those of you who have no clue, Wiki him. Unfortunately on mathgenealogy he has no lineage. Basically self-taught, he brought the story of solvability of quintics to an end. But to give credit to the right person, Abel solved the problem first. But Abel’s life is far less romantic than Galois. There was no duel, no French revolution, no early death at 21, no love affair, and most importantly, Abel wasn’t as good looking.
I don’t know how exactly I got into studying abstract algebra. In fact many chinese students seem to come out of college math degree without much of a background in AA. And that’s understandable. Only the handful of number theorists and representation theorists seem to rely excessively heavily on a deep understsanding of AA, whereas the rest of us Newtonian followers find analysis already hard enough.
So lately I encountered an interesting forum problem: construct the vertices of a square with only compass, and no straight edge. For those of you who are mathematically less mature (but still above average), that means you must construct the vertices by a series of intersections of circles each of which is centered at a previously constructed point and with radius given by the distance between two previously constructed points. Now nobody I am aware of has solved this problem. I tried for a few hours with a bunch of false solutions but no real progress. It leads us to believe this might be impossible, much like the ancient Greek problems of squares a circle or trisecting an angle, except there straight edge is allowed. The problem will be trivial if we allow straight edge here, for starters. So to prove such a bold statement, one conceivably needs to invoke Galois theoretical machinary. But contrary to my early childhood fantasy, it’s not that once you learned the basic tools of Galois theory you prove inconstructibility of anything that is indeed inconstructible. There are still numerous tricks involved, much like the case of quintics. And that’s indeed how math differs from many other scientific and engineering disciplines. Knowing the right framework to work in isn’t half the battle yet. It’s only a starting point.