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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 ¦µ 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 ¡Ý 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