伽羅瓦理論

前言

　　There are too many errors in the first edition， and so a "corrected nth printing" would have been appropriate. However， given the opportunity to makechanges， I felt that a second edition would give me the flexibility to changeany portion of the text that I felt I could improve. The first edition aimedto give a geodesic path to the Fundamental Theorem of Galois Theory，and I still think its brevity is valuable. Alas， the book is now a bit longer，but I feel that the changes are worthwhile. I began by rewriting almost allthe text， trying to make proofs clearer， and often giving more details thanbefore. Since many students find the road to the Fundamental Theoreman intricate one， the book now begins with a short section on symmetrygroups of polygons in the plane; an analogy of polygons and their symmetry groups with polynomials and their Galois groups can serve as a guideby helping readers organize the various definitions and constructions. Theexposition has been reorganized so that the discussion of solvability byradicals now appears later; this makes the proof of the Abel-Ruffini theorem easier to digest. I have also included several theorems not in the firstedition. For example， the Casus Irreducibilis is now proved， in keepingwith a historical interest lurking in these pages.　　I am indebted to Gareth Jones at the University of Southampton who，after having taught a course with the first edition as text， sent me a detailed list of errata along with perspicacious comments and suggestions. Ialso thank Evan Houston， Adam Lewenberg， and Jack Shamash who madevaluable comments as well. This new edition owes much to the generosityof these readers， and I am grateful to them.

內(nèi)容概要

本書是第二版，較第一版有很大的改進。證明更加清晰、詳盡。由于多變形對稱群和多項式的Galois群的相似性，書中以平面上的多邊形對稱群為開始。這種相似性可以幫助讀者理解書中的有關(guān)理論知識。書中也包含了一些新的定理，例如：不可約情形。書中用完整的證明和大量練習(xí)清晰、有效地講述了Galois理論。包括：立方、四次方公式的Galois理論的基本理論；五次Galois大定理的不可解性；立方和四次方Galois群的計算。補充了群論、尺規(guī)結(jié)構(gòu)和Galois的早期歷史。本書是一本Galois理論簡明教程，很適合研究生一年級作為教材學(xué)習(xí)；也是一本很理想的課外學(xué)習(xí)書。目次：對稱；環(huán)；同態(tài)和理想；商環(huán)；域上的多項式環(huán)；素理想和最大理想；不可約多項式；經(jīng)典多項式；分裂域；Galois群；單位根；根式可解性；特征的獨立性；Galois擴張；Galois理論的基本定理；應(yīng)用；Galois大定理；判別式；二次、三次、四次多項式的Galois群；結(jié)尾。

書籍目錄

Preface to the Second Edition Preface to the First Edition To the Reader Symmetry Rings Domains and Fields Homomorphisms and Ideals Quotient Rings Polynomial Rings over Fields Prime Ideals and Maximal Ideals Irreducible Polynomials Classical Formulas Splitting Fields The Galois Group Roots of Unity Solvability by Radicals Independence of Characters Galois Extensions The Fundamental Theorem of Galois Theory

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