## Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Google eBook)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. |

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### Contents

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 | |

### Common terms and phrases

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 Fermat's theorem fifth power follows form a2 Gauss given divisor gives implies infinite descent integer mod Kummer matrix mod h(a mod l3 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 integer primitive root 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

### Popular passages

Page vi - The sacred writers,' he observed, 'related the vicious as well as the virtuous actions of men ; which had this moral effect, that it kept mankind from despair, into which otherwise they would naturally fall, were they not supported by the recollection that others had offended like themselves...

Page 2 - On the other hand, it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as a sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers. I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.

Page 403 - Arithmeticorum libri sex et de numeris multangulis liber unus. Cum commentariis CG BACHETI et observationibus DP DE FERMAT.