**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2013; 200 pp; Softcover

MSC: Primary 11;

**Print ISBN: 978-0-8218-9848-2**

Product Code: MMONO/243

Product Code: MMONO/243

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

**Electronic ISBN: 978-1-4704-1627-0
Product Code: MMONO/243.E**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

#### This item is also available as part of a set:

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#### Supplemental Materials

# Fermat’s Last Theorem: Basic Tools

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*Takeshi Saito*

This book, together with the companion volume,
Fermat's Last Theorem: The proof, presents in full detail the
proof of Fermat's Last Theorem given by Wiles and Taylor. With these
two books, the reader will be able to see the whole picture of the
proof to appreciate one of the deepest achievements in the history of
mathematics.

Crucial arguments, including the so-called
\(3\)–\(5\) trick, \(R=T\) theorem, etc.,
are explained in depth. The proof relies on basic background materials
in number theory and arithmetic geometry, such as elliptic curves,
modular forms, Galois representations, deformation rings, modular
curves over the integer rings, Galois cohomology, etc. The first four
topics are crucial for the proof of Fermat's Last Theorem; they are
also very important as tools in studying various other problems in
modern algebraic number theory. The remaining topics will be treated
in the second book to be published in the same series in 2014. In
order to facilitate understanding the intricate proof, an outline of
the whole argument is described in the first preliminary chapter, and
more details are summarized in later chapters.

#### Readership

Graduate students and research mathematicians interested in number theory and arithmetic geometry.

#### Reviews & Endorsements

This book can serve as an introduction to the world of modularity results and will prove valuable for anyone willing to invest some work --- which of course one has to do in order to understand interesting mathematics. In the opinion of the reviewer, the author found a good balance between unavoidable omissions and desirable contents of a book like this.

-- Zentralblatt fur Mathematik

#### Table of Contents

# Table of Contents

## Fermat's Last Theorem: Basic Tools

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Preface to the English edition xv16
- Synopsis 118
- Elliptic curves 1330
- Modular forms 3552
- Galois representations 8198
- The 3–5 trick 111128
- 𝑅=𝑇 119136
- Commutative algebra 143160
- Deformation rings 159176
- Appendix A 171188
- Bibliography 189206
- Symbol index 197214
- Subject index 199216
- Back Cover Back Cover1218