Towers of Normal Subgroups

When we left off last week, I told you that I thought my way of understanding the quintic equation was different from what you'll find anywhere else. Today I want to elaborate on that.

When I took third-year algebra many years ago, the proof of the unsolvability of the quintic came about three-quarters of the way through the course, and it was just another corollory of a lemma in a long chain of proofs and lemmas that began with something about a "tower of normal subgroups" that had no physical or pictorial motivation. Later, when I asked the prof why you couldn't solve the quintic, she asked me: which line of the proof did you not understand?

That's not what math is to me. When I "understand" something in math, it's not because I've examined the proof line by line and verified that there are no flaws...it means on some level I actually understand it. I tried to explain this to the prof, but she just waved me off dismissively, as though my idea of understanding was just kid's stuff...that real mathematicians worked with logic and rigor and didn't waste their time with airy-fairy ideas about "understanding" things.

Well, just because I had one prof that didn't "get it", does that make me special? Of course not. But over the years I've encountered this attitude repeatedly. It's part of the culture of university professors to take that kind of attitude, and they're awfully smug about it. But I've never bought into it. That's whenever they say there's no "intuitive" way to understand these things, I'm determined to prove the opposite.

The other day I wrote about functions of the roots which map to each other under permutations of the roots. Those correspond to the "normal subgroups" they talk about in Galois theory...or, more correctly, they correspond to the quotient groups which are generated by those normal subgroups. The problem with the way Galois theory is presented is that they stick exclusively to those abstract concepts and never make them concrete by actually showing you those functions that I laid out for you in my last post. If you doubt me, search the internet and see if you can find anyone else talking about things like AB + CD mapping to AC + BD when you swap B and C. You won't find it. But that's the essence of the Galois group of the fourth degree equation.

To show how different my approach is from the traditional, we just have to compare the way Herstein treats the question. I used to have a copy of Herstein, but over the years it's gone astray. (It was actually Bill Leslie's copy...sorry about that, Bill!) If you read Wikipedia, you'll see that Herstein's text was considered the definitive undergraduate treatment for many years. More importantly, Herstein was noted (according to Wikipedia) for his clarity and knack for making things understandable.

As I said, I don't have it in front of me, but I remember a few things about it that stood out. Like every other authority, he's pretty clear that since the fifth degree is unsolvable, it's a waste of time trying to understand why by working your way up through the third and fourth, which are solvable. Then, to emphasize the pointlessness of trying to find sense in the pattern, he writes out the full (and horrifying) solution of the third degree in terms of the coefficients, concluding by saying that the fourth degree is even uglier.

This is completely opposite to my approach, which is to write out the solutions of the third and fourth not in terms of their coefficients, but implicitly in terms of symmetric functions of the roots. And I do it not to show how complicated and ugly they are, but how simple and beautiful.

From everything I've read, there is no one out there who tries to explain these things the way I do (although I give some credit to one Fiona Brunk for her clear and insightful historical narrative on an older website of hers). Oddly enough Herstein hails from Winnipeg (as I do); and in that connection I ought to mention one more ex-Winnipeger, the old-school Jewish genius Robert Israel whose name will be familiar to frequenters of internet math discussion groups, who graduated from St. John's high school just a few years ahead of me, and whose beautiful sister Susan was mooned over longingly by my older brother when we were all in junior high school together. (Aside to my wonderful sister-in-law: give it a rest, Donna! it was forty years ago!)  In this discussion on stackexchange.com, Robert says he doesn't think there is a "dumbed-down simple explanation".

Maybe not, depending on how "dumb" you expect. But I still think that's no excuse to throw up your hands and reject the idea of understanding these things altogether.