Algebraic Number Theory (Cambridge Studies in Advanced Mathematics Book 27) by A. Fröhlich (PDF)

Description

This book originates from graduate courses given in Cambridge and London. It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas. Throughout, the authors emphasise the systematic development of techniques for the explicit calculation of the basic invariants, such as rings of integers, class groups, and units. Moreover they combine, at each stage of development, theory with explicit computations and applications, and provide motivation in terms of classical number-theoretic problems. A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations. This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content. It will be indispensable for all practising and would-be algebraic number theorists.

User’s Reviews

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

⭐I have little to say except 5 stars, I just wanted to add to the reviews on this book since at present it has just one review.i’m not an expert but I think it’s a very clear book, and the authors have succeeded in their objective stated in the introduction, to provide a basic yet fairly ambitious treatment of first principles. My only complaint it the unusual numbering of statements, but I don’t mind a little variation from the norm sometimes, all authors should have liberty to arrange things as they see fit.This is a book about Dedekind domains. Reading this book will provide a grounding in the theory of Dedekind domains and provides a clear picture of how number fields are a special sort. It will also provide an introduction to the basic ideas of more advanced subjects like class field theory, such valuations and padic completions, Ostrowski’s theorem, the Dirichlet Unit Theorem and the Kummer criterion. Another feature is sections on not only quadratic and cyclotomic extensions, but less often treated cubic and biquadratic ones. There is very little homological algebra, so depending on preferences readers may enjoy having that ingredient hidden until (if and when) some real motivation has arisen.I think would be a good book for students of any level who have seen some algebra and wish to see how algebraic notions are applied.

⭐Algebraic Number Theory delivers. Maths texts are tough to review; this one performs as needed.

⭐It is an unfortunate feature of number theory that few of the books explain clearly the motivation for much of the technology introduced. Similarly, half of this book is spent proving properties of Dedekind domains before we see much motivation.That said, there are quite a few examples, as well as some concrete and enlightening exercises (in the back of the book, separated by chapter). There is also a chapter, if the reader is patient enough for it, on Diophantine equations, which gives a good sense of what all this is good for.The perspective of the book is global. Central themes are the calculation of the class number and unit group. The finiteness of the class number and Dirichlet’s Unit Theorem are both proved. L-functions are also introduced in the final chapter.While the instructor should add more motivation earlier, the book is appropriate for a graduate course in number theory, for students who already know, for instance, the classification of finitely generated modules over a PID. It may be better than others, but would be difficult to use for self-study without additional background.

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