有限场及伽罗瓦环讲义LECTURES ON FINITE FIELDS AND GALOIS RINGS

This is a textbook for graduate and upper level undergraduate students in mathematics, computer science, communication engineering and other fields. The explicit construction of finite fields and the computation in finite fields are emphasised. In particular, the construction of irreducible polynomials and the normal basis of finite fields are included. The essentials of Galois rings are also presented. This invaluable book has been written in a friendly style, so that lecturers can easily use it as a text and students can use it for self-study. A great number of exercises have been incorporated. --This text refers to the Hardcover edition.

作者简介：
　　Born in Shandong, China, in 1927, Zhe-Xian Wan graduated from Tsinghua University, Beijing, in 1948 and became a teacher at the same university. In 1950 he joined the Chinese Academy of Sciences, and in 1978 he was appointed Research Professor at the Academy. He is also a member of the Chinese Academy of Sciences.
　　Professor Wan's main research interests are algebra (pure and applied), combinatorics and coding theory. In addition to nearly 130 papers, he has published 18 books in these areas, including Classical Groups (jointly with L K Hua), Lie Algebras, Algebra and Coding, Introduction to Kac-Moody Algebras, Geometry of Classical Groups over Finite Fields, Geometry of Matrices and Quaternary Codes.

1 Sets and Integers
1.1 Sets and Maps
1.2 The Factorization of Integers
1.3 Equivalence Relation and Partition
1.4 Exercises
2 Groups
2.1 The Concept of a Group and Examples
2.2 Subgroups and Cosets
2.3 Cyclic Groups
2.4 Exercises
3 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 Exercises
4 Polynomials
4.1 Polynomial Rings
4.2 Division Algorithm
4.3 Euclidean Algorithm
4.4 Unique Factorization of Polynomials
4.5 Exercises
5 Residue Class Rings
5.1 Residue Class Rings
5.2 Examples
5.3 Residue Class Fields
5.4 More Examples
5.5 Exercises
6 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 Exercises
7 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 Exercises
8 Bases
8.1 Bases and Polynomial Bases
8.2 Dual Bases
……
9　Factoring Polynomials over Finite Fields
10　Irreducible Polynomials over Finite Fields
11　Quadratic Forms over Finite Fields
12　More Group Theory and Ring Theory
13　Hensel's Lemma and Hensel Lift
14　Galois Rings
Bibliography
Index