Gilbert J., Gilbert L. Elements of Modern Algebra

5th edition. — Brooks Cole, 1999. — 416 pages.The authors gradually introduce and develop concepts to help make the material more accessible. This text is intended for the introductory course in algebraic structures and covers groups before rings. This course often is used to bridge the gap from manipulative to theoretical mathematics and to help prepare secondary mathematics teachers for their careers. This text includes some optional sections to give instructors flexibility.
A minimal amount of mathematical maturity is assumed in the text; a major goal is to develop mathematical maturity. The material is presented in a theorem-proof format, with definitions and major results easily located thanks to a user-friendly format. The treatment is rigorous and self-contained, in keeping with the objectives of training the student in the techniques of algebra and providing a bridge to higher-level mathematical courses.Fundamentals.
Sets.
Mappings.
Properties of Composite Mappings (Optional).
Binary Operations.
Matrices.
Relations.
Key Words and Phrases
A Pioneer in Mathematics: Arthur Cayley
The Integers.
Postulates for the Integers (Optional).
Mathematical Induction.
Divisibility.
Prime Factors and Greatest Common Divisor.
Congruence of Integers.
Congruence Classes.
Introduction to Coding Theory (Optional).
Introduction to Cryptography (Optional).
Key Words and Phrases
A Pioneer in Mathematics: Blaise Pascal
Groups.
Definition of a Group.
Subgroups. Cyclic Groups.
Isomorphisms.
Homomorphisms.
Key Words and Phrases
A Pioneer in Mathematics: Niels Henrik Abel
More on Groups.
Finite Permutation Groups.
Cayley's Theorem.
Permutation Groups in Science and Art (Optional).
Normal Subgroups.
Quotient Groups.
Direct Sums (Optional).
Some Results on Finite Abelian Groups (Optional).
Key Words and Phrases
A Pioneer in Mathematics: Augustin Louis Cauchy
Rings, Integral Domains, and Fields.
Definition of a Ring.
Integral Domains and Fields.
The Field of Quotients of an Integral Domain.
Ordered Integral Domains.
Key Words and Phrases
A Pioneer in Mathematics: Richard Dedekind
More on Rings.
Ideals and Quotient Rings.
Ring Homomorphisms.
The Characteristic of a Ring.
Maximal Ideals (Optional).
Key Words and Phrases
A Pioneer in Mathematics: Amalie Emmy Noether
Real and Complex Numbers.
The Field of Real Numbers.
Complex Numbers and Quaternions.
De Moivre's Theorem and Roots of Complex Numbers.
Key Words and Phrases
A Pioneer in Mathematics: William Rowan Hamilton
Polynomials.
Polynomials over a Ring.
Divisibility and Greatest Common Divisor.
Factorization in F[x].
Zeros of a Polynomial.
Algebraic Extensions of a Field.
Key Words and Phrases
A Pioneer in Mathematics: Carl Friedrich Gauss
Appendix: The Basics of Logic
Answers to True/False and Selected
Exercises

Index