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
3.1.3 Self-reciprocal polynomials
3.1.4 Compositions of powers
3.1.5 Translation invariant polynomials
3.1.6 Normal replicators
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
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