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
2bxy Algebraic binary quadratic forms character mod Chinese remainder theorem class number formula coefficients computation condition congruence congruent mod conjugate corresponding cube cyclotomic integer g(a defined denote determinant Diophantus Dirichlet Disquisitiones Arithmeticae divides h(a divisible divisor class group Euler product Exercise exponent fact Fermat's Last Theorem fifth power finite follows form a2 Gauss given divisor gives implies infinite descent integer mod Kummer's matrix mod h(a modp modulo multiplicity exactly natural numbers Ng(a Nh(a nonzero norm number theory odd prime Pell's equation periods of length polynomial positive integer possible preceding section prime divisor prime factors prime integers primitive root primitive root mod principal divisor problem proof properly equivalent Pythagorean triple quadratic integers quadratic reciprocity quotient relatively prime remains prime representative set satisfies shown shows solution splitting class square mod squarefree suffice to prove values x,y integers zero