Fermat's Last Theorem: A Genetic Introduction to Algebraic Number TheorySpringer Science & Business Media, 14 janv. 2000 - 407 pages 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. |
Table des matières
Fermat | 1 |
Euler | 39 |
From Euler to Kummer | 59 |
Kummers theory of ideal factors | 76 |
Fermats Last Theorem for regular primes | 152 |
Determination of the class number | 181 |
Divisor theory for quadratic integers | 245 |
Gausss theory of binary quadratic forms | 305 |
Dirichlets class number formula | 342 |
The natural numbers | 372 |
Answers to exercises | 381 |
| 403 | |
| 409 | |
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Expressions et termes fréquents
a₁ Algebraic ax² binary quadratic forms character mod class number formula coefficients computation condition congruence congruent mod conjugate corresponding cube cy² cyclic method cyclotomic integer g(a cyclotomic integers defined denote determinant Diophantus Dirichlet Disquisitiones Arithmeticae divides divisible divisor class group equation Exercise exponent fact Fermat's Last Theorem fifth power finite follows g₁(a Gauss gives implies infinite descent integer mod Kummer mod h(a modulo n₁ n₂ natural numbers Ng(a nonzero norm number theory odd prime p₁ Pell's equation periods of length polynomial positive integer preceding section prime divisor prime factors primitive root mod principal divisors problem proof properly equivalent Pythagorean triple quadratic integers quadratic reciprocity quotient relatively prime remainder remains prime satisfies shown shows solution splitting class square mod squarefree suffice to prove u+vVD u₁ values x+yVD zero
Fréquemment cités
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