## 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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absolute value algebraic already analysis assume ball ball of radius called Cauchy sequence Chapter choose clear clearly closed ball coefficients compact complete compute condition consider construct contains continuous converges course defined definition divide divisible easy elements equal equation equivalent exactly example exists expansion extension fact factors field finite follows formal function given gives hence Hensel's Lemma implies inequality infinite integer interesting limit look means modulo multiplication non-archimedean norm Notice obtained p-adic polynomial positive power series prime Problem proof properties Proposition prove radius of convergence rational reader real number respect result ring roots of unity satisfies says sequence slope solution space Suppose tends theorem theory things true unique unit usual valuation vector write zero