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Title: A Course in Arithmetic (Graduate Texts in Mathematics) by Jean Pierre Serre ISBN: 0-387-90040-3 Publisher: Springer-Verlag Pub. Date: 01 March, 1973 Format: Hardcover Volumes: 1 List Price(USD): $54.95 |
Average Customer Rating: 5 (3 reviews)
Rating: 5
Summary: Breathtaking
Comment: Serre's work could best be summarized in one word - Elegance.
The book comprises of two distinct parts.
The first one is the 'algebraic' part. Serre's goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries to reaching this goal, he introduces the reader to quadratic reciprocity, p-adic fields and the Hilbert Symbol. After these three, he spends the next chapter detailing the properties of quadratic forms over Q and Q_p (the p-adic field). The reason to work over Q_p is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if and only if it has solutions in Q_p). Using Hensels Lemma, checking for solutions in Q_p is (almost) as easy as checking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note: the classification goal is already achieved in previous chapter).
The second half of the book is the 'analytic' one. The first chapter in this section gives a complete proof of Dirichlet's theorem while the second one studies the properties of modular forms (these are good!)
Due to the extreme elegance, the book is sometimes hard to read. This might sound like a paradox, but it's not and I'll explain why. The book takes some effort to read because it's terse and it often takes a while to figure out why something is 'obvious'. However, once you see it all, you'll realize that a great mind was guiding you through the pursuit. The choice of topics is just right to achieve the goals that the author sets out for himself. Also, I'd rather think for myself and read a smaller book than be given a huge fat tome where the author details his own thought process.
This book was my first foray into number theory and I absolutely enjoyed it. If you're considering reading it, I wish you joy in your pursuits.
Rating: 5
Summary: Very Demanding
Comment: The book is divided into two parts -- algebraic and analytic. I've only worked through the analytic part. Anything by Serre is worth its weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre's style is extremely condensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can't digest more than half a page of Serre a day; however if one wants to understand the structure of a theory, Serre is ideal.
I worked through "A Course in Arithmetic" over a decade back. As I recall I covered Riemann's zeta function and the Prime Number Theorem, the proof of Dirichlet's theorem on primes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modular functions. All of this material is also covered in Apostol's two books on analytic number theory ("Introduction to Analytic Number Theory", and "Dirichlet Series and Modular Functions in Number Theory"); Apostol goes further than Serre in the analytic part -- which is only to be expected since he is devoting two whole texts to the subject.
Rating: 5
Summary: A sheer delight
Comment: This short book on number theory by one of the giants of 20th century mathematics is delightful to read. Its length motivates one to finish the book, and it is packed full of interesting results. Most of the theory discussed in the book has wide-ranging applications, such as cryptography and dynamical systems. The last chapter of the book is the best and covers the subject of modular forms, including theta functions, Hecke operators, and general modular functions. If you want to understand the Wiles proof of Fermat's last theorem, start with this book.
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Title: The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics, 106) by Joseph H. Silverman ISBN: 0387962034 Publisher: Springer-Verlag Pub. Date: 01 January, 1986 List Price(USD): $64.95 |
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Title: Introduction to Elliptic Curves and Modular Forms (Graduate Texts in Mathematics, Vol 97) by Neal Koblitz ISBN: 0387979662 Publisher: Springer-Verlag Pub. Date: 01 May, 1993 List Price(USD): $62.95 |
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Title: Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics) by Tom M. Apostol ISBN: 0387901639 Publisher: Springer-Verlag Pub. Date: 01 June, 1976 List Price(USD): $64.95 |
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Title: Riemann's Zeta Function by Harold M. Edwards ISBN: 0486417409 Publisher: Dover Publications Pub. Date: 01 June, 2001 List Price(USD): $14.95 |
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Title: The Distribution of Prime Numbers (Cambridge Mathematical Library) by A.E. Ingham ISBN: 0521397898 Publisher: Cambridge University Press Pub. Date: 01 November, 1990 List Price(USD): $19.00 |
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