Finite Fields and Galois Rings

The following sections are included:

• Preface

• Contents

The following sections are included:

• Sets and Maps

• The Factorization of Integers

• Equivalence Relation and Partition

• Exercises

The following sections are included:

• The Concept of a Group and Examples

• Subgroups and Cosets

• Cyclic Groups

• Exercises

The following sections are included:

• Fields

• The Characteristic of a Field

• Rings and Integral Domains

• Field of Fractions of an Integral Domain

• Divisibility in a Ring

• Exercises

The following sections are included:

• Polynomial Rings

• Division Algorithm

• Euclidean Algorithm

• Unique Factorization of Polynomials

• Exercises

The following sections are included:

• Residue Class Rings

• Examples

• Residue Class Fields

• More Examples

• Exercises

The following sections are included:

• The Multiplicative Group of a Finite Field

• The Number of Elements in a Finite Field

• Existence of Finite Field with pⁿ Elements

• Uniqueness of Finite Field with pⁿ Elements

• Subfields of Finite Fields

• A Distinction between Finite Fields of Characteristic 2 and Not 2

• Exercises

The following sections are included:

• Automorphisms

• Characteristic Polynomials and Minimal Polynomials

• Primitive Polynomials

• Trace and Norm

• Quadratic Equations

• Exercises

The following sections are included:

• Bases and Polynomial Bases

• Dual Bases

• Self-dual Bases

• Normal Bases

• Optimal Normal Bases

• Exercises

The following sections are included:

• Factoring Polynomials over Finite Fields

• Factorization of xⁿ − 1

• Cyclotomic Polynomials

• The Period of a Polynomial

• Exercises

The following sections are included:

• On the Determination of Irreducible Polynomials

• Irreducibility Criterion of Binomials

• Some Irreducible Trinomials

• Compositions of Polynomials

• Recursive Constructions

• Composed Product and Sum of Polynomials

• Irreducible Polynomials of Any Degree

• Exercises

The following sections are included:

• Quadratic Forms over Finite Fields of Characteristic not 2

• Alternate Forms over Finite Fields

• Quadratic Forms over Finite Fields of Characteristic 2

• Exercises

The following sections are included:

• Homomorphisms of Groups, Normal Subgroups and Factor Groups

• Direct Product Decomposition of Groups

• Some Ring Theory

• Modules

• Exercises

The following sections are included:

• The Polynomial Ring ℤ_p^s[x]

• Hensel's Lemma

• Factorization of Monic Polynomials in ℤ_p^s[x]

• Basic Irreducible Polynomials and Hensel Lift

• Exercises

The following sections are included:

• Examples of Galois Rings

• Structure of Galois Rings

• The p-adic Representation

• The Group of Units of a Galois Ring

• Extension of Galois Rings

• Automorphisms of Galois Rings

• Generalized Trace and Norm

• Exercises

The following sections are included:

• Bibliography

• Index