Evariste Galois and His Theory

Like any math groupie with any pretensions to self-respect, I have long been somewhat familiar with the story of Evariste Galois. It's hard to imagine that there could be a more romantic tale in mathematics - Galois making his great discoveries as a teenager, frustrated in his attempts to get his work published, and dead in a duel at twenty - over a woman, no less (or maybe politics).

I have also been vaguely familiar with the content of his theory: he was the first to use the mathematical terminology group, played a key role in the development of group theory, and developed the eponymous Galois theory that established the impossibility of solution by radicals of the general polynomial equation of degree five or higher.

Despite having taken a couple of elementary courses in modern algebra, I found my teachers managed to stop before they actually got to Galois theory. Dean, for example, makes it the subject of his final chapter, and ditto the classic text Birkoff and MacLane. Dummit and Foote wait until page 538.

These facts, together with the failure of Cauchy and Fourier to appreciate his theory, convinced me of it's great complexity, and casual glances at the aforementioned texts did nothing to disabuse me.

One of the many delights of Robert Gilmore's new book, Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists is the initial chapter, which by way of motivation for the development of Lie Group theory throws in an elementary and concrete development of Galois Theory, shows how it permits the construction of solutions for degree four and below, and disallows general solutions for degree five and above. As a bonus, he shows how the theory resolves the age old puzzles about duplicating the cube, squaring the circle, and trisecting an angle. It takes him 22 pages, and doesn't use any math that shouldn't be familiar to college freshmen.