I have an abstract algebra exam coming up, and so I’ve been looking over everything we’ve covered so far. The subject and I haven’t really gotten along this past year. In particular, unlike in physics courses (where things are driven by the ‘big picture’), the point of the subject material was often quickly obscured. John Baez puts it perfectly:
“I used to hate this subject: it seemed like a massive waste of time. Newspapers, magazines and even lots of math books seem to celebrate the idea of people slaving away for centuries on puzzles whose only virtue is that they’re easy to state but hard to solve. For example: are any odd numbers the sum of all their divisors? Are there infinitely many pairs of primes that differ by 2? Is every even number bigger than 2 a sum of two primes? Are there any positive integer solutions to
for
? My response to all these was: who cares?!
Sure, it’s noble to seek knowledge for its own sake. But working on a math problem just because it’s hard is like trying to drill a hole in a concrete wall with your nose, just to prove you can! If you succeed, I’ll be impressed - but I’ll still wonder why you didn’t put all that energy into something more interesting.
Now my attitude has changed, because I’m beginning to see that behind these silly hard problems there lurks an actual theory, full of deep ideas and interesting links to other branches of mathematics, including mathematical physics. It just so happens that now and then this theory happens to crack another hard nut.”
Indeed: in particular, now that we’ve reached the section on Galois theory, I’ve developed a new appreciation for the subject (and the way it fits into everything else). The connection to physics is still somewhat fuzzy to me; but intuitively, it’s a beautiful way of looking at things. In particular, Baez’s post on the subject (where the above quote comes from) is a very nice introduction, as well as the first reference by Stewart that he cites.
2 responses so far ↓
blank // June 8, 2007 at 9:39 am
When I first came across Galois theory, I was really impressed to see how extremely simple ideas coalesce to form a profound theory of extreme importance.
It was sublime….
Robert // October 6, 2007 at 7:34 am
Galois theory is particularly great mathematics, and as an abstract algebra professor, I’m always disappointed that we almost always run out of semester before we can touch on this stuff. And like the previous commenter said, one of the prettiest things about Galois theory is that it’s familiar on the one hand — using ideas most people see in high school Algebra II — and then it goes to the outermost theory with those simple ideas. I think it’s very important to realize that even the most common things around us retain an element of mystery to them.
There are some nice applications to cryptography, too, not just physics.
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