域和伽羅瓦理論

內(nèi)容概要

域是有理數(shù)集合、實數(shù)集合、復(fù)數(shù)集合的抽象模型，因此在整個數(shù)學(xué)科學(xué)中處于基礎(chǔ)地位。Galois是最早提出有限域觀點的人，他對于抽象域理論的誕生至關(guān)重要。 本書把抽象域論一分為二，首先講代數(shù)擴張及其在代數(shù)域論上的應(yīng)用，其次介紹超越擴張及其在代數(shù)函數(shù)論及代數(shù)幾何上的應(yīng)用，中間還插入經(jīng)典的Galois理論，使讀者對于實際背景有比較清楚的認(rèn)識。

書籍目錄

Preface Notes to the Reader List of Symbols I Galois Theory   1 Field Extensions   2 Automorphisms   3 Normal Extensions   4 Separable and Inseparable Extensions   5 The Fundamental Theorem of Galois Theory II Some Galois Extensions   6 Finite Fields   7 Cyclotomic Extensions   8 Norms and Traces   9 Cyclic Extensions   10 Hilbert Theorem 90 and Group Cohomology   11 Kummer Extensions III Applications of Galois Theory   12 Discriminants   13 Polynomials of Degree 3 and 4   14 The Transcendence of and e   15 Ruler and Compass Consturctions  16 Solvability by RadicalsⅣ Infinite Algebraic ExtensionsⅤ Transcendental ExtensionsAppendix A Ring TheoryAppendix B Set TheoryAppendix C Group TheoryAppendix D Vector SpacesAppendix E TopologyReferencesIndex

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