有限域和伽罗华环讲义
2006-11
世界图书出版公司北京公司
Zhe-Xian Wan
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本书是为数学专业，计算机科学专业，以及通信工程等专业的研究生和高年级大学生编写的教本。书中强调了具体构造有限域以及限域中的计算，其中包括有限域上多项式的分解，不可约多项规基，以及伽罗华环的基本内容。本书除了用作教材还可以用来自学。书中还有大量习题。
1 Sets and Integers 1.1 Sets and Maps 1.2 The Factorization of Integers 1.3 Equivalence Relation and Partition 1.4 Exercises2 Groups 2.1 The Concept of a Group and Examples 2.2 Subgroups and Cosets 2.3 Cyclic Groups 2.4 Exercises3 Fields and Rings 3.1 Fields 3.2 The Characteristic of a Field 3.3 Rings and Integral Domains 3.4 Field of Fractions of an Integral Domain 3.5 Divisibility in a Ring 3.6 Exercises4 Polynomials 4.1 Polynomial Rings 4.2 Division Algorithm 4.3 Euclidean Algorithm 4.4 Unique Factorization of Polynomials 4.5 Exercises5 Residue Class Rings 5.1 Residue Class Rings 5.2 Examples 5.3 Residue Class Fields 5.4 More Examples 5.5 Exercises6 Structure of Finite Fields 6.1 The Multiplicative Group of a Finite Field 6.2 The Number of Elements in a Finite Field 6.3 Existence of Finite Field with pn Elements 6.4 Uniqueness of Finite Field with pn Elements 6.5 Subfields of Finite Fields 6.6 A Distinction between Finite Fields of Characteristic 2 and Not 2 6.7 Exercises7 Further Properties of Finite Fields 7.1 Automorphisms 7.2 Characteristic Polynomials and Minimal Polynomials 7.3 Primitive Polynomials 7.4 Trace and Norm 7.5 Quadratic Equations 7.6 Exercises8 Bases 8.1 Bases and Polynomial Bases 8.2 Dual Bases……9 Factoring Polynomails over Finite Fields10 Irreducible Polymials over Finite Fields11 Quadratic Forms over Finite Fields12 More Group Theory and Ring Theory13 Hensel's Lemma and Hensel Lift14 Galois RingsBibliographyIndex
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