## 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. |

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algebraic closure analysis anXn archimedean ball of radius Cauchy sequence closed ball coefficients complete with respect compute condition congruent consider contains Corollary course defined definition divisible easy elements equal equation equivalent example exists extension of Qp fact factors finite extension follows gi(X gives hence Hensel's Lemma Hint inequality infinite irreducible polynomial isomorphism l)-st roots Let f(X logarithm metric minimal polynomial modulo monic multiplication Newton polygon non-archimedean absolute value non-zero norm Notice number theory open ball p-adic absolute value p-adic analysis p-adic expansion p-adic numbers polynomial f(X positive integer positive real number power series prime 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(x Weierstrass Preparation Theorem ZP[X