I studied some algebraic number theory as part of my Masters project, and proved a series of exercises that lead to the proof of what is know as the 'Kronecker-Weber theorem'.

It was a wonderful morning, in fact, it hardly dawned on me that it was morning when I went to sleep!!! So I got myself ready, and cycled to the department. It was 6:45 am, this was the first time that I reached the department so early (I have gone back to my room at this time from the department many a times though :P).

And then did the last minute preparations- check my slides, read my notes and blah blah blah.

My talk was scheduled at 10:10 am. The department had played it safe to ensure that there was a sizable audience, in that it had promised a delicious lunch to all those who attended the talks.

I made my last minute calls back home(with the number of exams that I have to answer reducing drastically, I do not miss an opportunity to let folks at home wish me 'best of luck').

So finally the moment arrived. I motioned to the screen, and began the jargon...

I mentioned before, I spoke on algebraic number theory. Here is some part of the my introduction -

The subject developed due to a rather silly comment of mathematician Pierre Fermat in the margin of Arithmetica by Diophantus as follows -

*I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.*

*Fermat claimed that he had an elegant proof to prove the above fact. Unfortunately he was dead before one could publish books with fatter margins!!! The search for this coveted proof has lasted lifetimes of some of the best mathematicians in the past. This was notoriously labeled as the 'Fermats last theorem'(FLT). It was proved in 1994 by Prof Andrew Wiles at Princeton university. The proof is hardly elementary, it is about 290 pages long and uses some of the most sophisticated modern algebraic number theory. It definitely needs a lot more than a margin of a book.*

However the alarming simplicity of the statement generated the most number of 'wrong proofs' for this problem in the history of mathematics( my attempt included-my undergraduate professor almost fainted when I told him that I was trying to prove the 'Fermats last theorem'!!!). In fact it was this theorem that in a way got me exited about mathematics. Very often it is difficult to prove some of the most simple questions in our lives. By simple I mean, ones with really simple statements, so simple that even a five year kid could understand them. Mathematics is flooded with such questions, and the FLT is a prime example.

The basic reason why many of the earlier proofs collapsed for due to the assumption of unique factorization in certain domains. Mathematics in general, and number theory in particular has a predominant inductive reasoning, in that I mean that, one proves something for something small and finite, and then extrapolates it to something much larger. A similar reasoning was applied to the set of integers. Integers have this marvelous property called the 'Fundamental theorem of arithmetic', which asserts that every integer can be written as a unique product of prime powers. One should note that this theorem(like many other useful theorems) has a dual nature, that is it not only talks about an existence of factorization but also guarantees its uniqueness. Uniqueness is a very important property, if something can be done uniquely, it means that it is independent of who does it. It also helps one to come up with important formulas(by computing something in two different ways and then equating them). This general principle was assumed for domains containing the integers, and this assumption was wrong. The moral of the story is that whilst proving something, check that the assumptions are proved to be correct-otherwise you might end up with absolute non-sense.

Algebraic number theory was developed to answer questions which generalize properties of integers to bigger and general domains... from here on the talk becomes a little technical and hence cannot write it up here...

I have still got to write out the report though, so will pushoff for now, till then all the best...may be you could start writing out conjectures in your textbooks, who know we might soon have a 'last theorem' after your name!!!