{"product_id":"algebraic-number-theory-and-fermats-la","title":"Algebraic Number Theory And Fermats La","description":"\u003cp\u003eUpdated to reflect current research and extended to cover more advanced topics as well as the basics, \u003cb\u003eAlgebraic Number Theory and Fermat’s Last Theorem, Fifth Edition\u003c\/b\u003e introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers, initially from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work.\u003cbr\u003e \u003cbr\u003e \u003cb\u003eNew to the Fifth Edition\u003c\/b\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ePell''s Equation x^2-dy^2=1: all solutions can be obtained from a single `fundamental'' solution, which can be found using continued fractions.\u003c\/li\u003e\n\u003cli\u003eGalois theory of number field extensions, relating the field structure to that of the group of automorphisms.\u003c\/li\u003e\n\u003cli\u003eMore material on cyclotomic fields, and some results on cubic fields.\u003c\/li\u003e\n\u003cli\u003eAdvanced properties of prime ideals, including the valuation of a fractional ideal relative to a prime ideal, localisation at a prime ideal, and discrete valuation rings.\u003c\/li\u003e\n\u003cli\u003eRamification theory, which discusses how a prime ideal factorises when the number field is extended to a larger one.\u003c\/li\u003e\n\u003cli\u003eA short proof of the Quadratic Reciprocity Law based on properties of cyclotomic fields. This\u003c\/li\u003e\n\u003cli\u003eValuations and \u003ci\u003ep\u003c\/i\u003e-adic numbers. Topology of the \u003ci\u003ep\u003c\/i\u003e-adic integers.\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eWritten by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.\u003c\/p\u003e","brand":"MediaPlace","offers":[{"title":"Default Title","offer_id":57195963941246,"sku":"NW9781032610931","price":48.9,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1379\/1261\/files\/9781032610931.jpg?v=1778540003","url":"https:\/\/mediaplace.com\/products\/algebraic-number-theory-and-fermats-la","provider":"MediaPlace","version":"1.0","type":"link"}