Fermat's Last Theorem A Genetic Introduction to Algebraic Number Theory [Hardcover]

This introduction to algebraic number theory via the famous problem of Fermats Last Theorem follows its historical development, beginning with the work of Fermat and ending with Kummers theory of ideal factorization. The more elementary topics, such as Eulers proof of the impossibilty of x+y=z, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummers theory to quadratic integers and relates this to Gauss'theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.Work on this book was supported in part by the James M. Vaughn, Jr., Vaughn Foundation Fund.This book is an introduction to algebraic number theory via the famous problem of Fermat's Last Theorem. The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of ideal factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.1 Fermat.- 1.1 Fermat and his Last Theorem. Statement of the theorem. History of its discovery..- 1.2 Pythagorean triangles. Pythagorean triples known to the Babylonians 1000 years before Pythagoras..- 1.3 How to find Pythagorean triples. Method based on the fact that the product of two relatively prime numbers can be a square only if both factors are squares..- 1.4 The method of infinite descent..- 1.5 The casen= 4 of the Last Theorem. lÃ]