## Wednesday, August 22, 2007

### Fermat's Last Theorem, Germain and Galois

A small introduction. The curse of slow reading has fallen upon me and I must struggle with it. However, through this struggle I finished reading Fermat's Last Theorem, by Simon Singh a while ago. This post hasn't been written straight after I finished reading the book, since I wanted to see how well I remember it. Call it a test of my character! (Or to see whether or not slow reading is pointless.)

For 'completeness' Fermat's Last Theorem states that,

$x^n +y^n =z^n$

has no non-zero integer solutions for x, y and z when n > 2.

This is a generalisation of Pythagoras's Theorem I believe, and I hadn't really been aware of it that much until reading the book mentioned above. I have come across and used Fermat's Little Theorem, one which I grew to like, but whenever the Last Theorem was mentioned it didn't register in my empty head.

Upon reading that book, I surprisingly feel mathematically informed. I think that is a must read book for any mathematics student. A definite must read. It informed me of the history of Mathematics, which is fascinating in its own right. About Alexandria, Pythagoras and lots of fascinating mathematicians. Simon Singh has brilliantly outlined, not only the history of Fermat's Last Theorem, but the lives of the mathematicians who lived and struggled with that theorem.

The natural choice is to begin with Fermat and his 'ickle' theorem. Fermat famously stated in his copy of Artithmetica, 'I have discovered a truly remarkable proof which this margin is too small to contain.'

A word of warning if I may - don't ever write 'the margin is too small' for an answer on an exam question! You could try it on your homeworks and hope your supervisor has a sense of humour, but don't risk it in exams. [If I recall correctly, Hardy was afraid of travelling by sea and so used to send the message that he had solved the Riemann Hypothesis(?) before setting sail as an insurance policy. His logic being that God wouldn't want for him to drown with that knowledge...)

I know many people believe that Fermat didn't actually have the proof, and must surely have made a mistake, but I believe otherwise. It is strange how we form these conclusions, but I'm certain - having read about him - that he did have the proof. Fermat might have annoyed his fellow mathematicians, but he seems like a Mourinho type of guy! (In a weird way of course). Now this book, as well as enlightening us about Fermat's Theorem told the story of how mathematicians contributed to it. A lot of mathematicians did contribute towards it and a lot of maths was discovered in the process. Apart from Euler, the two names that had a big impact on me whilst reading have to be Évariste Galois (of course), and Sophie Germain. I say Euler because 'Infinite Descent' (the proposed name for the magazine) came about because of his proof for when n=3. Euler was indeed a remarkable mathematician, but I think I 'grew to hate his name' in college, when we solved differential equations by his method.

\begin{aside}
I think part of the maths syllabus should incorporate the history of maths. It doesn't have to be examinable, but if maths students do read about it, they will be amazed and might also become more interested in mathematics. I mean, having read about Euler properly in a different context, I think he was a cool guy! The history of maths puts maths in an entirely different light and mathematicians as well. For others who don't see it as an alive subject, the history of it gives maths a life. A heart beat. When the programme Dangerous Knowledge had been on TV the other day, surprisingly watched by Noddy, I was told that what had been interesting was the life of the mathematicians (and how they were freaks, if I may add!) Hehe, I think this 'Noddy' needs to be neutralised!

However the point is that Noddy found the history of maths and the history of mathematicians interesting. I know that it hasn't 'inspired' Noddy to look at maths in a different light (just given evidence of how I may turn out to be, which is why I discouraged my parents from watching it. ;) ). But that was probably the first time that I've discussed maths with Noddy, rather than spoken to myself about it.
\end{aside}

Back to the 'big three'.** Germain stood out for different reasons. During her time women studying maths was frowned upon and universities were only for men. However, thankfully Germain found a way around it; albeit under the guise of being a male! I think the most striking thing was Germain's determination to study mathematics, regardless of her circumstances. Germain corresponded with the mathematical world, after her parents finally relented and let her study maths, by pretending to be a French male student. Her parents had initially restricted her from studying maths and taken away her candles and ink, but still she had persevered. I'll dodgily skip the bit in the middle, and jump to the bit when she decided to communicate her ideas on Fermat's theorem with Gauss. Or should I say 'he decided to'... ? The communication (and friendship) between the two resulted in Gauss's life being saved by Germain and her true identity being revealed.

Another thing that struck me about Germain, was that when Gauss moved onto astronomy and stopped replying to her letters, she felt 'different' and left pure mathematics. Her inspiration and (role model) Gauss, probably didn't realise what he gave to Germain in terms of motivation. I know there are some people who don't need this inspiration, but as someone who needs to be inspired and motivated most times, I can understand what losing that inspiration feels like. You really can't be bothered with anything for a while, and unless you find a different source or rediscover some back up inspiration, you struggle. However, although Germain had lost her motivation for pure mathematics, she still chugged on!

Now finally there is Galois. His story has got to be the one that has most affected me and had a positive impact on me. Embarrassingly for me, during the year when I had been talking to a lecturer, Galois Theory had been mentioned. I had no idea who Galois was at that time and had never heard of him, so had stupidly come away thinking that the lecturer had meant Galileo! The weird thing is that when I said his name to someone else, they assumed the same thing as me! (Don't worry - I soon corrected my ridiculous assumption).

Galois's story is a sad one. Situations and circumstances were against him throughout his life, which was riddled with misfortune. One example of why political involvement is bad is Galois. OK, times have changed, (correct me if I'm wrong) but political turmoil was happening in France at that time. Galois may have been 'hot headed', but I can sympathise with him. Feeling helpless at times compels you to act in an irrational manner. I'm not saying that it is good to do so, but Galois was a kid and he went through so much! One reason why I probably was taken in by his story, was the fact that when he had been 19, already so much misfortunes had been witnessed. He was 20 I think, when he died - one year older than me!

His genius can't be denied and he'd only studied maths for a short time. Maybe if he had filled in the gaps to his thoughts and calculations, he might have got into the university he wanted? Because he did a lot of working out in his head, he left others who did follow his thoughts, confused. Much of the mathematics that he discovered is not understood by me. I haven't even done Groups rigorously yet, but it all sounds very interesting. (I do understand the basic concept of a group though, but only had two proper lectures in December on this).

I have probably gone on, and could still go on, but as you've guessed this story doesn't have a happy ending. Galois was murdered, and I suppose we can all speculate as to why this happened. I think that the 'powers that be' sent a woman to do the job. Galois fell for this woman and her 'to be' husband found out. (I think he was one of the top guns in the government and wasn't pleased as you can imagine, upon discovering this). Now he was probably a convincing actor in my opinion, and had planned for this to happen. He challenged Galois to a duel and there was only meant to be one winner. :( It wasn't Galois if you're interested.

Indeed there has been less of the actual theorem and Andrew Wiles massive, gigantic, enormous, power infinity(!) determination to solve it mentioned here! However, the book will enlighten you about all that. The book beautifully constructed the journey to the proof, and it is impossible not to be taken in by events. Galois was actually my final inspiration and probably the most important one, to type my proposals up. I had been lazy, as per usual, but when voting commenced on the name, I was back in the drivers seat. The Galois Group might have two meanings, but to me there is a third. That of a young mathematician who had to endure hardships, and not study maths further. (Although it does show massive strength of character to go about heading a rebellion!) The third meaning is that what we take from his story. He sounded like a cool kid as well!

To conclude (yes there is one), I would definitely recommend the book. It might have some concepts unfamiliar to A Level students etc, but read it is as a story book. Read about the links between different mathematicians and mathematics like a story, for that is what it is. This maths story is not an infinite one, but it will continue for a long time yet. You don't have to read the beginning to follow it, but it definitely changes your perspective of things having read from the beginning. And as I've written elsewhere; it is indeed mathematicians of the past that still motivate maths today. (Oh, and if you haven't noticed, there has been an obvious bias towards Galois in this post!)

*Will add links etc \sout{tomorrow} after some sleep! Since it's been a while I've posted a long post, I can't estimate how long this is, which will also be dealt with after some sleep (and painting).
** United are no longer part of the top 3. Humbug.