Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory
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.
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From Euler to Kummer
Kummers theory of ideal factors
Fermats Last Theorem for regular primes
Determination of the class number
Divisor theory for quadratic integers
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Algebraic applied argument assume assumption binary quadratic forms called Chapter character class number coefficients complex computation condition congruence conjugate consider contains Conversely corresponding course cube cyclic method cyclotomic integers defined definition denote determinant Dirichlet distinct divides divisible divisor class element equal equation equivalent Euler exactly example Exercise exist fact Fermat's Last Theorem follows form a2 formula Gauss given gives ideal implies impossible Kummer least less mod h(a multiplicity namely natural necessary norm Note occur periods positive integer possible preceding section prime divisor prime factors primitive principal problem proof prove quadratic integers question reduced relation relatively prime remains representation root satisfies shown shows side simple solution solved splitting square statement suffice Table theory true unique unit values written zero