| 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 Polynomails over Finite Fields 10 Irreducible Polymials 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 |
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