Failure, the birth pangs of success

I'm teaching a class called Modern Algebra. Sometimes it's called Abstract Algebra. It's a required course for math majors, and USM has quite a few more of those than my old employer did, so it's full. More than that, it way over the cap. Several graduating seniors appealed to my mercy, and I relented.

One way of looking at Modern Algebra is that it's an example of how an incredible failure in mathematics can nevertheless produce stunningly beautiful tools useful in ways that no one imagined from the original problem.

If you've taken high school algebra, then you should remember that you learned how to solve linear equations such as

x + 2 = 5.

You also probably learned how to solve quadratic equations such as

x² + 2x + 5 = 0.

The solution can always be found using the quadratic formula, an example of "solving by radicals." The ancients knew how to solve such equations using another method that actually produces the quadratic formula, but the formula itself wasn't discovered untili the Middle Ages in India (or so I read).

Moving on, you almost certainly never learned how to solve cubic equations such as

x³ + 2x + 5 = 0.

There is, however, a cubic formula that solves this equation by radicals, just as there is a quadratic formula that solves the quadratic equation by radicals. The reason you didn't learn it is that it's rather complex, and no one in his right mind trusts the average high school student with such things. Even math majors don't often see it.

The cubic formula was discovered during the Renaissance by an Italian named Tartaglia, and published by another Italian named Cardano, to whom Tartaglia had shown the formula under an oath of secrecy. Cardano got around the oath by hiring a youngster named Ferrari to solve it, and giving him a hint. That led to a lawsuit, which I won't get into here. Ferrari also discovered a quartic formula that solves quartic equations by radicals, such as

x⁴ + 2x + 5 = 0

but that's even more complicated than the cubic formula.

Well, what of quintic equations such as

x⁵ + 2x + 5 = 0?

Here mathematicians hit a three century-long speed bump. It took a series of discoveries by three mathematicians—among others, Paolo Ruffini, Evariste Galois, and Niels Abel—to explain the delay. It turns out that

equations of degree 5 and higher do not admit a general formula for solution by radicals.

Curiously, Tartaglia and Cardano had the afore-mentioned lawsuit, Ferrari was poisoned, and Ruffini's proof was incomplete, while Galois and Abel both died young. Galois died in a duel, apparently over a woman, and his death may have been assured by the fact that he stayed up the night before in order to commit to paper his ideas. Mathematicians sometimes lead interesting lives.

While progressing doggedly towards this spectacular failure, mathematicians developed a number of extremely useful tools which abstracted properties of polynomials and their roots. (Hence "abstract" algebra.) Despite the fact that these tools failed to provide a general formula by radicals, and to the contrary demonstrated that no such formula existed, they nevertheless have proven extremely useful in many other situations, including problems that have nothing to do with polynomials. Group theory, for example, was used to predict and describe the existence of quarks before they were verified by experiment. Another example would be the security of your online credit card transactions.

So this is material I have always found fascinating. Even grading papers has been more interesting than in the past. I try to convey the beauty of the material to the students, but sometimes I dwell too long on certain technical details that probably make them lose sight of the big picture. Gotta work on that; perhaps a few sketches of proofs would be preferable to all the wretched details.