Thursday, February 23, 2006

Non-Fiction - "The Equation That Couldn't be Solved"

This book, written by Mario Livio, is a short history of group theory and the two main mathematicians who created it. These two guys are Niels Henrik Abel and Evariste Galois. Both of these mathematicians died very young. Abel at 26 from tuberculosis and Galois died at 20 from a pistol shot in a duel.

The book begins with the history of the problem that led to the invention of group theory. The problem was finding a formula that used only basic arithmetical operations and radicals (i.e. taking of roots) to write a general solution to an equation. We are all familiar with the quadratic formula. Similar, although more complicated, formulas exist for the cubic and quartic (i.e. degree 3 and degree 4) equations. However, no one was able to come up with such a formula for the quintic (degree 5).

It turns out that there is a very good reason why no one found such a formula. It does not exist. This fact was proved both by Abel independently by Galois. In the process they laid the foundation for modern abstract algebra - group theory in particular.

The life story of both of these man is rather sad - although some were able to appreciate their genius - they were both unrecognized during their lifetime. Throughout his short life Abel was unable to secure permanent employment and he died dirt poor.

Galois was passionate about politics and in early 19th centuary France this got him into all kinds of trouble - he was expelled from school and spent some time in jail. It is not entirely clear why Galois accepted a challenge to a duel. However, he was sure he was going to die because the night before he worked hard to organize his papers and write down his most important ideas.

Both Abel and Galois had the bad luck of submitting their papers to the French Academy (which was the center of math research at the time) and the mathematicians assigned to review them just lost them. These were guys like Fourier and Cauchy. In the days before copy machines this was a pretty serious setback.

The historical and biographical chapters were the best part of this book. The math was unsatisfying because the author would stop just as the arguments became interesting. I suppose he was trying to avoid having too many equations in the text.

The last part of the book talked about applications of group theory. In particular group theory is used in places where symetry occurs - and the author comes up with tons of examples from art and sciences. One well known finite group is the group composed of the face rotations of a Rubik's Cube.

But I found this talk about symetries little dull and shallow. I want to find out more about Galois's and Abel's proof. I have bunch of books on the subject but I will have to brush up on abstract algebra.


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