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　　分類: 图书,进口原版书,科学与技术 Science & Techology ,

作者： Antoine Chambert-Loir著

出 版 社：

出版时间： 2004-10-1字数：版次： 1页数： 195印刷时间： 2004/10/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387214283包装： 精装内容简介

This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendence of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibility theorem. The reader will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. In explaining these concepts, the author also provides comments on their historical development and leads the reader along many interesting paths.

In addition, there are theorems from analysis: as stated before, the transcendence of the numbers pi and e, the fact that the complex numbers form an algebraically closed field, and also Puiseux's theorem that shows how one can parametrize the roots of polynomial equations, the coefficients of which are allowed to vary. There are exercises at the end of each chapter, varying in degree from easy to difficult. To make the book more lively, the author has incorporated pictures from the history of mathematics, including scans of mathematical stamps and pictures of mathematicians.

作者简介

Antoine Chambert-Loir taught this book when he was Professor at École Polytechnique, Palaiseau, France. He is now Professor at Université de Rennes 1.

目录

1 Field extensions

Constructions with ruler and compass

Fields

Field extensions

Some classical impossibilities

Symmetric functions

Appendix: Transcendence of e and π

Exercises

2 Roots

Ring of remainders

Splitting extensions

Algebraically closed fields; algebraic closure

Appendix: Structure of polynomial rings

Appendix: Quotient rings

Appendix: Puiseux's theorem

Exercises

3 Galois theory

Homomorphisms of an extension in an algebraic closure

Automorphism group of an extension

The Galois group as a permutation group

Discriminant; resolvent polynomials

Finite fields

Exercises

4 A bit of group theory

Groups (quick review of basic definitions)

Subgroups

Group actions

Normal subgroups; quotient groups

Solvable groups; nilpotent groups

Symmetric and alternating groups

Matrix groups

Exercises

5 Applications

Constructibility with ruler and compass

Cyclotomy

Composite extensions

Cyclic extensions

Equations with degrees up to 4

Solving equations by radicals

How (not) to compute Galois groups

Specializing Galois groups

Hilbert's irreducibility theorem

Exercises

6 Algebraic theory of differential equations

Differential fields

Differential extensions; construction of derivations

Differential equations

Picard-Vessiot extensions

The differential Galois group; examples

The differential Galois correspondence

Integration in finite terms, elementary extensions

Appendix: Hilbert's Nullstellensatz

Exercises

Examination problems

References

Index

 

 

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