book: Number Theory 1: Fermat's Dream (Translations of Mathematical Monographs)

book: Number Theory 1: Fermat's Dream (Translations of Mathematical Monographs) | Kazuya Kato, Nobushige Kurokawa, ... ...

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	• Number Theory 1: Fermat's Dream (Translations of Mathematical Monographs) Kazuya Kato, Nobushige Kurokawa, ... American Mathematical Society, 2000 - 154 pages average customer review:based on 1 review view larger image
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An accessible path from numbers to the sophisticated methods

This brilliant précis of algebraic number theory goes from the simplest basics up to suggestive outline proofs of the most important theorems. It is probably not a good first book on the subject although it is pretty self-contained in principle. You should already be comfortable with groups, rings, and fields.

Its great strength is connecting easily stated questions about Diophantine equations to more sophisticated methods. Chapter 0 quickly relates Pell's equations x^2+Ny^2=1 to square and triangular numbers, and both to the groups of units of algebraic number rings. Chapter 1 on elliptic curves is illustrated throughout by a handful of typical examples, and it proves many steps, but not all, in the Mordell theorem: the rational points on an elliptic curve form a finitely generated Abelian group.

There are many insightful comments on how to think about the ideas overall. These rise to poetry when we learn that the p-adic numbers are like the night sky, beauty obscured when the "sun" of the rational numbers blots them out.

The high point, typifying the book's style, is the last chapter explaining how arithmetic in an algebraic number field K is captured by the kernel and cokernel of one group homomorphism (the group map that takes each unit of the field to its principal fractional ideal-kernel is the unit group and cokernel the class group). The remark is lightly made. Examples are given. But in fact the stage is being set for cohomological methods.

All this in 140 pages. Plus there are exercises, and answers to them.

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This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series.