- 有限域手册(英文)/国外优秀数学著作原版系列
- - 作者：编者:(美)盖里·L.马伦//(加)丹尼尔·帕纳里约|责编:刘立娟
  - 出版社：哈尔滨工业大学
  - ISBN：9787560398167
  - 出版日期：2022/05/01
  - 页数：1041
- 售价：71.2

内容大纲
本书主要介绍了有限域的理论和应用，通过有限域的历史和导论引入了其理论性质，介绍了不可约多项式、本原多项式等，还介绍了有限域上的一些性质，包括有限域上的特殊函数、有限域上的序列、有限域上的曲线等，阐述了代数编码理论、密码学及应用等内容，以及有限域在组合数学、代数编码理论、密码学系统、生物学、量子信息理论、工程以及其他领域的各种数学和实际应用。本书提供了一个综合的索引，包括3000多条文献，读者可以快速找到最新的有关于有限域的事实与结论。
作者介绍
目录
Part I: Introduction
  1  History of finite fields
    1.1  Finite fields in the 18-th and 19-th centuries  Roderick Gow
      1.1.1  Introduction
      1.1.2  Early anticipations of finite fields
      1.1.3  Gauss's Disquisitiones Arithmeticae
      1.1.4  Gauss's Disquisitiones Generales de Congruentiis
      1.1.5  Galois's Sur la theorie des nombres
      1.1.6  Serret's Cours d'algebre superieure
      1.1.7  Contributions of SchSnemann and Dedekind
      1.1.8  Moore's characterization of abstract finite fields
      1.1.9  Later developments
  2  Introduction to finite fields
    2.1  Basic properties of finite fields Gary L. Mullen and Daniel Panario
      2.1.1  Basic definitions
      2.1.2  Fundamental properties of finite fields
      2.1.3  Extension fields
      2.1.4  Trace and norm functions
      2.1.5  Bases
      2.1.6  Linearized polynomials
      2.1.7  Miscellaneous results
        2.1.7.1  The finite field polynomial Φ function
        2.1.7.2  Cyclotomic polynomials
        2.1.7.3  Lagrange interpolation
        2.1.7.4  Discriminants
        2.1.7.5  Jacobi logarithms
        2.1.7.6  Field-like structures
        2.1.7.7  Galois rings
      2.1.8  Finite field related books
        2.1.8.1  Textbooks
        2.1.8.2  Finite field theory
        2.1.8.3  Applications
        2.1.8.4  Algorithms
        2.1.8.5  Conference proceedings
    2.2  Tables  David Thomson
      2.2.1  Low-weight irreducible and primitive polynomials
      2.2.2  Low-complexity normal bases
        2.2.2.1  Exhaustive search for low complexity normal bases
        2.2.2.2  Minimum type of a Gauss period admitting a normal basis of F2n over F2
        2.2.2.3  Minimum-known complexity of a normal basis of F2n over F2, n ≥ 40
      2.2.3  Resources and standards
Part II: Theoretical Properties
  3  Irreducible polynomials
    3.1  Counting irreducible polynomials  Joseph L. Yucas
      3.1.1  Prescribed trace or norm
      3.1.2  Prescribed coefficients over the binary field

    3.2  Construction of irreducibles  Melsik Kyuregyan
      3.2.1  Construction by composition
      3.2.2  Recursive constructions
    3.3  Conditions for reducible polynomials  Daniel Panario
      3.3.1  Composite polynomials
      3.3.2  Swan-type theorems
    3.4  Weights of irreducible polynomials  Omran Ahmadi
      3.4.1  Basic definitions
      3.4.2  Existence results
      3.4.3  Conjectures
    3.5  Prescribed coefficients  Stephen D. Cohen
      3.5.1  One prescribed coefficient
      3.5.2  Prescribed trace and norm
      3.5.3  More prescribed coefficients
      3.5.4  Further exact expressions
    3.6  Multivariate polynomials  Xiang-dong Hou
      3.6.1  Counting formulas
      3.6.2  Asymptotic formulas
      3.6.3  Results for the vector degree
      3.6.4  Indecomposable polynomials and irreducible polynomials
      3.6.5  Algorithms for the gcd of multivariate polynomials
  4  Primitive polynomials
    4.1  Introduction to primitive polynomials  Gary L. Mullen and Daniel Panario
    4.2  Prescribed coefficients  Stephen D. Cohen
      4.2.1  Approaches to results on prescribed coefficients
      4.2.2  Existence theorems for primitive polynomials
      4.2.3  Existence theorems for primitive normal polynomials
    4.3  Weights of primitive polynomials  Stephen D. Cohen
    4.4  Elements of high order  Jose Felipe Voloch
      4.4.1  Elements of high order from elements of small orders
      4.4.2  Gao's construction and a generalization
      4.4.3  Iterative constructions
  5  Bases
    5.1  Duality theory of bases  Dieter Jungnickel
      5.1.1  Dual bases
      5.1.2  Self-dual bases
      5.1.3  Weakly self-dual bases
      5.1.4  Binary bases with small excess
      5.1.5  Almost weakly self-dual bases
      5.1.6  Connections to hardware design
  ……
Part Ⅲ: Applications
Bibliography
Index
编辑手记

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