| Number Theory III: Diophantine Geometry 1991 Edition Contributor(s): Lang, Serge (Author), Lang, Serge (Editor) |
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ISBN: 3540530045 ISBN-13: 9783540530046 Publisher: Springer OUR PRICE: $104.49 Product Type: Hardcover Published: June 1991 |
| Additional Information |
| BISAC Categories: - Mathematics | Number Theory - Mathematics | Geometry - Algebraic |
| Dewey: 512.7 |
| LCCN: 91017718 |
| Series: Encyclopaedia of Mathematical Sciences |
| Physical Information: 0.75" H x 6.14" W x 9.21" (1.36 lbs) 296 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out- standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em- phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in- sights. Fermat's last theorem occupies an intermediate position. Al- though it is not proved, it is not an isolated problem any more. |