Thursday 25 June 2015

Math Girls; Galois theory

It is never easy to teach someone else, and it is even harder to teach others an advanced topic. In university the ideal case would be giving the motivation during the lecture and students do the rest by themselves during self-study or supervision. But, if we really ought to teach someone from the very beginning, what would be the best choice? This is of course a very complicated question and for sure I cannot give the answer here.

My way to do it is to follow the historical treatment - doing what those mathematicians did hundreds of years ago. What they did, why they did so --- this is exactly what helps students to understand the mechanism behind a topic. Landau styled definition-theorem-proof aren't bad, they are just a bit too hard for those understanding ability not being the best.

Based on the above I have found a series interesting, the Math Girls by H.Yuki. There are 5 vol. available in Chinese and Japanese, and the first 3 vol. are published in English. Such series perfectly illustrates the above. Topics delivered via conversation and story-telling, and by merging yourself into the discussion the history behind will simply push you all the way through to the end of the journey.

It is mostly easy for senior secondary school students --- mostly refers to the most part of the book. It starts basically from stretch and the difficulty gradually increase, but it is usually tolerable till the last chapter. Nonetheless one will be able to appreciate the theory without precisely understanding the technical procedures.

A few months back I've just finishing reading his 5th book of the series on Galois theory. One of the clear motivation behind the theory is the irresolvability of quintic equations in radicals. This is of course a very hard question, that had puzzled mathematicians for three hundred years, until a genius called Galois came up with his genius idea (that no one can realize till some years later) a day before his duel, where he lost and died. It was a sad story (he is one of the many French mathematicians that died in weird ways), but this is not the main point --- where was his idea came from?

It is the permutation of roots.

Classical algebra books will illustrate this via quadratic equations, cubic equations and quadric equations. However this is not very clear under numerical examples beyond the quadratic case. In particular, Cardano's solution for cubic equation is fairly unpleasant. There are so many radicals added --- which one gives an extension? Which one does not? What is the degree of extension?... When I first tried to read through graduate texts on Galois theory I kept asking questions like this to myself and got myself puzzled.

In the book they put some effort analyzing those equations in an elemental way. No groups, no fields --- this is also what mathematicians did in the past. I personally found that extremely useful by the end of the day, and I loved that book.

It might be a bit more approachable using modern language than Galois' first thesis, but it does not change the fact that his thesis is showing the core idea of the whole theory, so following the historical treatment it would be the best for one to follow the thesis to explore the elemental part of his theory. That was not obvious until the very last bit [Kummer extensions, main theorem for quintic equations], so I will try to fill that gap here, if I have time, probably in the coming entry.

And yeah, for those who are interested in mathematics [in particular if you know how to read Chinese or Japanese], I would sincerely recommend that to you.


Theorem. $x^5-16x + 2 = 0$ has no radical solution over $\mathbb{Q}$.


Friday 5 June 2015

An interesting sum

Here is a probability problem recently: Suppose you want to make a system that a player eventually wins --- this is pretty straightforward in the theoretical sense because under a sequence of independent trials with winning rate $p>0$ it is always possible to win eventually.

We may model the problem as a roulette problem, where we have $N$ balls inside a box in which one of them is special.

The simplest model is the geometric model --- we put the non-special ball back and retry the whole process. It follows a geometric distribution with expectation $\frac{1}{p}$. But this is far too slow, and it has a large variance with $p$ is small.