This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations.
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"...an excellent contribution to the long list of books presenting the main results of algebraic number theory. It is useful for anyone who is learning or teaching this branch of mathematics." J. Browkin, Mathematical Reviews
Book Description
The systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, is emphasized throughout this introduction to the foundations of algebraic number theory.
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7 of 9 people found the following review helpful:
4.0 out of 5 stars
Technical, but concrete,
By W. Calvert (Murray, Kentucky, USA) - See all my reviews
This review is from: Algebraic Number Theory (Cambridge Studies in Advanced Mathematics) (Hardcover)
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.
1 of 1 people found the following review helpful:
5.0 out of 5 stars
algebraic number theory,
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This review is from: Algebraic Number Theory (Cambridge Studies in Advanced Mathematics) (Paperback)
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.
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Inside This Book
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First Sentence:
In this chapter we seek to lay down the algebraic foundations which will be needed. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
discrete absolute value, uniformising parameter, finite commutative algebra, tower formula, imaginary embeddings, biquadratic fields, unique prime ideal, normalised basis, finite separable extension, primitive mth root, distinct prime ideals, decomposition law, quadratic subfields, distinct embeddings, quadratic reciprocity law, ordinary absolute value, valuation ideal, primitive idempotents, algebraic integers, inertia group, class number one, real embeddings, unique factorisation, principal ideal domain, integral closure Key Phrases - Capitalized Phrases (CAPs): (learn more)
Chinese Remainder Theorem, Hensel's Lemma, Dirichlet's Unit Theorem, Algebraic Theory, Blichfeldt's Lemma, Gauss Lemma New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this chapter we seek to lay down the algebraic foundations which will be needed. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
discrete absolute value, uniformising parameter, finite commutative algebra, tower formula, imaginary embeddings, biquadratic fields, unique prime ideal, normalised basis, finite separable extension, primitive mth root, distinct prime ideals, decomposition law, quadratic subfields, distinct embeddings, quadratic reciprocity law, ordinary absolute value, valuation ideal, primitive idempotents, algebraic integers, inertia group, class number one, real embeddings, unique factorisation, principal ideal domain, integral closure Key Phrases - Capitalized Phrases (CAPs): (learn more)
Chinese Remainder Theorem, Hensel's Lemma, Dirichlet's Unit Theorem, Algebraic Theory, Blichfeldt's Lemma, Gauss Lemma New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 18
books:
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100
books
cite this book:
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- Class Field Theory (Advanced Book Classics) by Emil Artin in Back Matter
- Local Fields (London Mathematical Society Student Texts) by J. W. S. Cassels in Back Matter
- Transcendental Number Theory (Cambridge Mathematical Library) by Alan Baker in Back Matter
- The Higher Arithmetic: An Introduction to the Theory of Numbers by H. Davenport in Back Matter
- Number Theory: An approach through history from Hammurapi to Legendre by André Weil in Back Matter
See all 18 books this book cites
- The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (v. 106) by Joseph H. Silverman in Back Matter (1), Back Matter (2), and Back Matter (3)
- Rational Homotopy Theory (Graduate Texts in Mathematics) by Yves Felix in Back Matter (1), and Back Matter (2)
- Complex Analysis (Graduate Texts in Mathematics) by Serge Lang in Front Matter, and Back Matter
- The Geometry of Schemes (Graduate Texts in Mathematics) by David Eisenbud in Back Matter (1), and Back Matter (2)
- Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics) by Harold M. Edwards in Back Matter (1), and Back Matter (2)
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