Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics, Vol. 50) (Graduate Texts in Mathematics, 50)

A virtuous mix of history, calculation, and inspiration.

This is a good book to read about mathematics. All of us have witnessed, at least the ones who have strong interests in math, how often mathematics amounts to little more than gamesmanship, and often trivial at that. Fermat's Last Theorem in this respect is a good case study, because the work on the theorem started out as little more than the typical game-playing, and it gradually grew beyond that to connect up with the great river of mathematics, right at its heart. Prof. Edwards discussion gives one an opportunity to trace this curious evolution and to gain some insights into mathematics and how, from humble beginnings, a mathematics problem that piques the interest of people can grow into something of perennial interest. I'm only about 1/3 of the way through the book, but have found it to be fascinating reading.

Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

Let a,b,c,n be positive whole numbers where n > 1c=a+bc^n =(a+b)^n = a^n + b^n + rem(a,b,n) by binomial expansionc^n = a^n + b^n iff rem(a,b,n)=0rem(a,b,n) >0c^n <> a^n + b^nQEDNow think about the equation of a circle... :)