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### Course Name: Introduction to finite field and their application 课程名称: 有限域理论与应用

Lecturer Professor: Zeng Jiwen

Text book: Introduction to finite fields and their application
Author: Rudolf Lidl, Harald Niederreiter
Publishing Company: Cambridge University
Published date: 1994

Contents:
1. Algebraic foundations: Groups, Rings and Fields, Polynomials, Field Extensions
2. Structure of Finite fields: Characterization of finite Fields, Roots of Irreducible polynomials, traces, Norms, Bases, Roots of Unity and Cyclotomic polynomials, Representation of elements of finite fields, Wedderburn’s Theorem
3. Polynomials and Primitive Polynomials, Irreducible polynomials, Constructiom of irreducible polynomials, Linearized polynomials, Binomials and Trinomials
4. Factorizationof Polynomials: Factorization over small finite fields, Factorizations over large finite fields, Calculation of Roots of polynomials,
5. Exponential Sums: Characters, Gaussian Sums
6. Linear Recurring Sequences: Feedback Shift Registers, Periodicity properties, Impulse response sequences, Generating functions, The minimal polynomials, Families of Linear recurring sequences,
7. Theoretical Applications of Finite fields: Finite geometries, Combinatorics, Linear modular systems, Pseudorandom sequences
8. Algebraic Coding theory: Linear codes, Cyclic codes, Goppa codes
9. Cryptology: Stream Ciphers, Discrete logarithms,

1 全英文板书, 中英文互补讲课
2 周学时3-4, 讲授一学期
3 每周布置一次练习, 要求英文解答
4 成绩判定标准: 考试60%, 平时练习30%, 上课率10%.
5 预备课程要求: 本课程需要学生有较好的线性代数背景。其它代数背景知识，如群，环，理想等可在课程中简单介绍，如需要时，再详细解释。本课程的特点是理论与应用并行，故被广泛用作教材. 本课程适合高年级的本科生选修，同时更适合研究生选修。