## p-adic Numbers: An IntroductionIn the course of their undergraduate careers, most mathematics majors see little beyond "standard mathematics:" basic real and complex analysis, ab stract algebra, some differential geometry, etc. There are few adventures in other territories, and few opportunities to visit some of the more exotic cor ners of mathematics. The goal of this book is to offer such an opportunity, by way of a visit to the p-adic universe. Such a visit offers a glimpse of a part of mathematics which is both important and fun, and which also is something of a meeting point between algebra and analysis. Over the last century, p-adic numbers and p-adic analysis have come to playa central role in modern number theory. This importance comes from the fact that they afford a natural and powerful language for talking about congruences between integers, and allow the use of methods borrowed from calculus and analysis for studying such problems. More recently, p-adic num bers have shown up in other areas of mathematics, and even in physics. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

1 | |

Foundations 23 | 22 |

padic Numbers | 43 |

Elementary Analysis in Qp 87 | 86 |

Vector Spaces and Field Extensions | 133 |

### Other editions - View all

### Common terms and phrases

algebraic closure analysis archimedean ball of radius Cauchy sequence closed ball coefficients complete with respect compute condition congruent consider contains Corollary course defined definition divisible easy equal equation equivalent exactly example exists extension of Qp fact factors finite extension follows gives hence Hensel's Lemma hi(X Hint inequality infinite irreducible polynomial isomorphism Let f(X logarithm means metric minimal polynomial modulo monic multiplication Newton polygon non-archimedean absolute value non-zero norm Notice number theory open ball open set p-adic absolute value p-adic analysis p-adic expansion p-adic numbers p"Zp polynomial f(X positive integer positive real number power series Problem proof properties Proposition prove quotient rational numbers reader region of convergence residue field ring roots of unity satisfies says series converges slope solution Suppose tends to zero topology unique unramified vector space vp(a Weierstrass Preparation Theorem Zp[X