Number Theory in Function Fields (Graduate Texts in Mathematics, 210) 2002nd Edition by Michael Rosen (PDF)

Description

Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.

User’s Reviews

Editorial Reviews: Review From the reviews:MATHEMATICAL REVIEWS”Both in the large (choice and arrangement of the material) and in the details (accuracy and completeness of proofs, quality of explanations and motivating remarks), the author did a marvelous job. His parallel treatment of topics…for both number and function fields demonstrates the strong interaction between the respective arithmetics, and allows for motivation on either side.”Bulletin of the AMS”… Which brings us to the book by Michael Rosen. In it, one has an excellent (and, to the author’s knowledge, unique) introduction to the global theory of function fields covering both the classical theory of Artin, Hasse, Weil and presenting an introduction to Drinfeld modules (in particular, the Carlitz module and its exponential). So the reader will find the basic material on function fields and their history (i.e., Weil differentials, the Riemann-Roch Theorem etc.) leading up to Bombieri’s proof of the Riemann hypothesis first established by Weil. In addition one finds chapters on Artin’s primitive root Conjecture for function fields, Brumer-Stark theory, the ABC Conjecture, results on class numbers and so on. Each chapter contains a list of illuminating exercises. Rosen’s book is perfect for graduate students, as well as other mathematicians, fascinated by the amazing similarities between number fields and function fields.”David Goss (Ohio State University) From the Back Cover Elementary number theory is concerned with arithmetic properties of the ring of integers. Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting, for example, analogues of the theorems of Fermat and Euler, Wilsons theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlets theorem on primes in an arithmetic progression. After presenting the required foundational material on function fields, the later chapters explore the analogy between global function fields and algebraic number fields. A variety of topics are presented, including: the ABC-conjecture, Artins conjecture on primitive roots, the Brumer-Stark conjecture, Drinfeld modules, class number formulae, and average value theorems.The first few chapters of this book are accessible to advanced undergraduates. The later chapters are designed for graduate students and professionals in mathematics and related fields who want to learn more about the very fruitful relationship between number theory in algebraic number fields and algebraic function fields. In this book many paths are set forth for future learning and exploration.Michael Rosen is Professor of Mathematics at Brown University, where hes been since 1962. He has published over 40 research papers and he is the co-author of A Classical Introduction to Modern Number Theory, with Kenneth Ireland. He received the Chauvenet Prize of the Mathematical Association of America in 1999 and the Philip J. Bray Teaching Award in 2001.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a dense and scholarly review ofa topic that lends itself very well tocomputation (cf. other books on the samesubject). Sadly the text is unleavened withillustrative (or even any!) examples to naildown the exposition. Too bad – a real missedopportunity.

⭐Excellent introduction to the subject. The beginning of the book is accessible to advanced undergraduates. The book emphasizes the algebraic viewpoint. Very well-written.

⭐Zusammen mit Ireland verfasste Rosen eines der Standartwerke zur algebraischen Zahlentheorie.In diesem Buch widmet er sich der Zahlentheorie über endlichen Funktionenkörpern. Dabei werden aus der “normalen” Zahlentheorie bekannte Theoreme in den Spezialfall der Funktionenkörper übersetzt, was vielerorts beim Verständnis hilfreich ist, da man den Allgemeinfall im Hinterkopf hat.Leider beschränkt er sich bei seinen Beweisen fast ausschließlich auf das allernotwendigste. Mit “short calculation” abgetane Begründungen können durchaus mehrere essenzielle Denkschritte erfordern und alles andere als “short” oder “easy” sein. Kleinere, aber doch häufiger auftauchende Tippfehler erschweren die Arbeit mit diesem Werk zusätzlich. Die Lektüre des Buches ist somit sehr langwierig, sofern man vor hat sich erstmalig auf diesem Gebiet zu bewegen.Fazit: ein gut strukturiertes Buch zu einem spannenden Thema, aber leider für den Einsteiger nur mit großer Mühe zu bewerkstelligen.Best book to start with function field

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Free Download Number Theory in Function Fields (Graduate Texts in Mathematics, 210) 2002nd Edition in PDF format
Number Theory in Function Fields (Graduate Texts in Mathematics, 210) 2002nd Edition PDF Free Download
Download Number Theory in Function Fields (Graduate Texts in Mathematics, 210) 2002nd Edition 2002 PDF Free
Number Theory in Function Fields (Graduate Texts in Mathematics, 210) 2002nd Edition 2002 PDF Free Download
Download Number Theory in Function Fields (Graduate Texts in Mathematics, 210) 2002nd Edition PDF
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