A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics, 84) 2nd Edition by Kenneth Ireland (PDF)

Description

This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves.

User’s Reviews

Editorial Reviews: Review From the reviews of the second edition:K. Ireland and M. RosenA Classical Introduction to Modern Number Theory”Many mathematicians of this generation have reached the frontiers of research without having a good sense of the history of their subject. In number theory this historical ignorance is being alleviated by a number of fine recent books. This work stands among them as a unique and valuable contribution.”― MATHEMATICAL REVIEWS”This is a great book, one that does exactly what it proposes to do, and does it well. For me, this is the go-to book whenever a student wants to do an advanced independent study project in number theory. … for a student who wants to get started on the subject and has taken a basic course on elementary number theory and the standard abstract algebra course, this is perfect.” (Fernando Q. Gouvêa, MathDL, January, 2006)

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

⭐I have a B.S. in mathematics and I always did well in my courses; I was particularly good at number theory. My undergraduate class used

⭐, which is actually a pretty great book. Looking for something more advanced, I signed up for an independent reading course, and this is the book the professor assigned.First of all, I do not recommend this text unless you have a strong background in algebra. Number theory and abstract algebra are inextricably linked, and this book makes frequent use of the connections, but without doing much to explain anything that more solidly falls under the “algebra” heading. Without a good understanding of field theory, this book will be beyond your grasp.This is, without a doubt, a “difficult” text. It’s very terse, and while the proofs are elegant, they’re often quite mysterious. I can’t even count the number of times that the phrase “It’s obvious that…” has left me completely mystified, and it’s a gleeful moment when I can pencil in the margin that it actually IS obvious, for once. The exercises are frequently more difficult than it seems the author’s intended; several of them have stumped my professor, and the motivation isn’t always obvious.This leads me to my main point: This is not a book for learning number theory for the first time! This isn’t even a book for learning number theory for the second time. This is a book for developing an extremely rigorous understanding of a complex subject once you already have a wide variety of tools at your disposal and already possess a solid foundation in mathematics.The difficulty level of the text isn’t the reason for the “low” review score. The typesetting is, in several places, ambiguous. The notation can lead to confusion in even interpreting an exercise or statement. This seems to be mostly a result of lack of effort; I don’t see a reason why the Legendre/Jacobi symbol can’t always be made easily distinguishable from regular division. Context should help make the distinction, but if you’re having a hard time understanding what’s going on, the added level of frustration in simply interpreting the notation is just superfluously discouraging.Essentially, this can be a challenging text to work through, and you’ll find very little in the way of support in its pages. I’ve found myself turning to other references countless times to get a handle on some of the results, and I think a lot of that explanation could easily have been included in the first place. I’m not a fan of “elegant” math in the learning process; I’m a fan of explanations, examples, and connections… all of which are in extremely short supply in this text.

⭐The selection of topics is very good, and the historical notes at chapter endings are interesting. Most of the text is a reprint of the first edition for which the authors received an extensive list of corrections. It is surprising, then, that there are so many typographical errors. At a detriment clarity there seems to be an effort to include a great deal of content in a small amount of space. For example, mathematical expressions containing fractions, exponents, subscripts, superscripts, and subscripts on exponents (which could gain clarity from proper display) are printed inline. Towards the end of a line these may be broken at inappropriate places. Sometime exponents of one expression are intermingled with subscripts of another expression on the previous line. The size of subscripts on exponents can challenge the printing process to distinguish ink blots from characters. Other space saving devices include abbreviating the hypotheses of propositions, and omitting some of the details of proofs,

⭐This book is a model of elegant and concise writing that is delightful to read … provided you have the necessary background. By that, I mean a familiarity with (abstract) algebra at the undergraduate level, and a level of mathematical “maturity”. The authors often provide proofs that are concise but clear. They demonstrate how, with a little algebra, we can acquire a deeper grasp of basic theorems like “Fermat’s Little Theorem” (which is just something that drops out as a corollary once the appropriate lemmas and theorems are proved), and concepts like primitive roots, etc.As far as coverage goes, it does not attempt a very comprehensive treatment of all the major topics in number theory. Thus, while multiplicative number theory is elegantly and insightfully treated, additive number theory is missing. Instead, the authors move from the foundations towards areas of current interest, such as elliptic curves. Perhaps that is why they call it “Modern Number Theory”. The reader who wishes to study some of the more classical aspects of number theory could consult other texts like Hardy & Wright, or Niven.

⭐I have worked through all the problems in the first 8 chapters, and return to it constantly-have read most of the book one way and another. The (very readable) writing style really enables a student to understand an underlying theme of ideas well. A truly beautiful selection of topics that have helped in my own research in writing papers in number theory (especially the end notes of the chapters). A book that I regard with great affection, and will always carry with me. I can never completely express my gratitude to the authors sufficiently. It just occurred to me that rather than take my word for it, read the introduction to the book “Gauss and Jacobi sums” by Berndt, Evans, and Williams, in which Prof. B. Berndt, and Prof. R. Evans, both experts in number theory, explicitly credit this particular book as being their inspiration. It is one of the great number theory textbooks around today.

⭐I am a self-studier so I thrive on texts that are self-contained, give beautiful proofs, and make eye-opening observations. (Not observations that add to perplexity.)This book sets the gold standard for all of those criteria.I bought this based on the glowing comments on two major, high-powered math forums – where the commentators were extremely well-versed in the material. They certainly knew what they were talking about.

⭐The book does require some background in algebra, but is a surprisingly easy readconsidering the amount of material that it covers. Despite being 25 years old, it’s agood introduction to many topics in number theory.

⭐Great book but arrived slightly damaged. A partial refund per compensation would be suitable.

⭐Pour tout étudiant License en maths quelle que soit sa spécialité ultérieure .good

⭐To be read so I cannot give a report but appears ok.

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