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Algebra: Pure and Applied
Aigli Papantonopoulou, The College of New Jersey

ISBN-10: 0130882542
ISBN-13: 9780130882547

Publisher: Prentice Hall
Copyright: 2002
Format: Paper; 550 pp
Published: 05/24/2001

Suggested retail price: $73.33
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For a one-semester course covering groups and rings or a two-semester course in Abstract Algebra.

This text provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of applications. A repetition and examples first approach introduces students to mathematical rigor and abstraction while teaching them the basic notions and results of modern algebra.

  • Concrete examples precede difficult concepts.

    • Builds students' intuition through experience with specific examples so that they may more readily absorb abstract concepts.

  • Repetition of key steps in proofs.

    • Gives students many opportunities to learn the difficult steps in proofs by first introducing them in examples, then in the actual proof of the theorem, and again in later examples.

  • Complete algebraic theorems and proofs.

    • Combines either a complete proof or a proof that students can complete with every algebraic theorem mentioned so that students do not have to reference outside sources.

  • Self-contained treatment of all main topics.

    • Introduces the topics essential to abstract algebra in such a way that students can follow the arguments of a proof without needing to turn to other works for help.

  • Unifying theme in each chapter.

    • Helps students draw conceptual parallels.

  • Solutions of cubics and quartics are given first.

    • Allows students to explore the connections between those solutions and geometric constructions and learn how their Galois groups are calculated.

  • A brief history of algebra is discussed in a separate chapter.

    • Allows for a more meaningful discussion of the evolution of the subject.

  • Applications such as Symmetries, Grobner bases, Coding Theory, and Boolean algebras are provided with references.

    • Gives students the opportunity to learn more on their own, and instructors flexibility in topical coverage.

  • Chapter 0—Covers basic material used in the development of the main theories of algebra.

    • Allows instructors to teach this material in an introductory course. Gives more advanced students a useful reference.

  • The interplay between algebra and geometry is emphasized.

    • Shows students how closely the two subjects are linked…ñespecially useful for future high school teachers who will be teaching both subjects.

  • 64 sets of exercises—Containing an average of 20 exercises each.

    • Gives students ample opportunity to practice the concepts they just learned.

  • A large number of worked out examples.

    • Illustrates for the student how an abstract concept is used before its formal definition is stated.



Preface.


Acknowledgments


0. Background.

Sets and Maps. Equivalence Relations and Partitions. Properties of Z. Complex Numbers. Matrices.

A. GROUP THEORY.

1. Groups.

Examples and Basic Concepts. Subgroups. Cyclic Groups. Permutations.

2. Group Homomorphisms.

Cosets and Lagrange's Theorem. Homomorphisms. Normal Subgroups. Quotient Groups. Automorphisms.

3. Direct Products and Abelian Groups.

Examples and Definitions. Computing Orders. Direct Sums. Fundamental Theorem of Finite Abelian Groups.

4. Group Actions.

Group Actions and Cayley's Theorem. Stabilizers and Orbits in a Group Action. Burnside's Theorem and Applications. Conjugacy Classes and the Class Equation. Conjugacy in Sn and Simplicity of A5. The Sylow Theorems. Applications of the Sylow Theorems.

5. Composition Series.

Isomorphism Theorems. The Jordan-Hölder Theorem. Solvable Groups.

B. RINGS AND FIELDS.

6. Rings.

Examples and Basic Concepts. Integral Domains. Fields.

7. Ring Homomorphisms.

Definitions and Basic Properties. Ideals. The Field of Quotients.

8. Rings of Polynomials.

Basic Concepts and Notation. The Division Algorithm in F[x]. More Applications of the Division Algorithm. Irreducible Polynomials. Cubic and Quartic Polynomials. Ideals in F[x]. Quotient Rings of F[x]. The Chinese Remainder Theorem for F[x].

9. Euclidean Domains.

Division Algorithms and Euclidean Domains. Unique Factorization Domains. Gaussian Integers.

10. Field Theory.

Vector Spaces. Algebraic Extensions. Splitting Fields. Finite Fields.

11. Geometric Constructions.

Constructible Real Numbers. Classical Problems. Constructions with Marked Ruler and Compass. Cubics and Quartics Revisited.

12. Galois Theory.

Galois Groups. The Fundamental Theorem of Galois Theory. Galois Groups of Polynomials. Geometric Constructions Revisited. Radical Extensions.

13. Historical Notes.

From Ahmes the Scribe to Omar Khayyam. From Gerolamo Cardano to C. F. Gauss. From Evariste Galois to Emmy Noether.

C. SELECTED TOPICS.

14. Symmetries.

Linear Transformations. Isometries. Symmetry Groups. Platonic Solids. Subgroups of the Special Orthogonal Group. Further Reading.

15. Grobner Bases.

Lexicographic Order. A Division Algorithm. Dickson's Lemma. The Hilbert Basis Theorem. Gröbner Bases and the Division Algorithm. Further Reading.

16. Coding Theory.

Linear Binary Codes. Error Correction and Coset Decoding. Standard Generator Matrices. The Syndrome Method. Cyclic Codes. Further Reading.

17. Boolean Algebras.

Lattices. Boolean Algebras. Circuits. Further Reading.

Answers and Hints to Selected Exercises.

Bibliography.

Index.

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