Introduction to Applied Algebraic Systems

This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as well as a fairly complete discussion of major algebraic systems (such as rings, fields, and groups) with a view of their use in bar coding, public key cryptosystems, error-correcting codes, counting techniques, and elliptic key cryptography. This is the only entry level text for algebraic systems that includes an extensive introduction to elliptic curves, a topic that has leaped to prominence due to its importance in the solution of Fermat's Last Theorem and its incorporation into the rapidly expanding applications of elliptic curve cryptography in smart cards. Computer science students will appreciate the strong emphasis on the theory of polynomials, algebraic geometry and Groebner bases. The combination of a rigorous introduction to abstract algebra with a thorough coverage of its applications makes this book truly unique.

Table of Contents:
1: Modular Arithmetic 1.1: Sets, Functions, Numbers 1.2: Induction 1.3: Divisibility 1.4: Prime Numbers 1.5: Relations and Partitions 1.6: Modular Arithmetic 1.7: Equations in Zn 1.8: Bar Codes 1.9: The Chinese Remainder Theorem 1.10: Euler's -function 1.11: Theorems of Euler and Fermat 1.12: Public Key Cryptosystems 2: Rings and Fields 2.1: Basic Properties 2.2: Subrings and Subfields 2.3: Review of Vector Spaces 2.4: Polynomials 2.5: Polynomial Evaluation and Interpolation 2.6: Irreducible Polynomials 2.7: Construction of Finite Fields 2.8: Extension Fields 2.9: Multiplication Structure of Finite Fields 2.10: Primitive Elements 2.11: Subfield Structure of Finite Fields 2.12: Minimal Polynomials 2.13: Isomorphisms Between Fields 2.14: Error Correcting Codes 3: Groups and Permutations 3.1: Basic Properties 3.2: Subgroups 3.3: Permutation Groups 3.4: Matrix Groups 3.5: Even and Odd Permutations 3.6: Cayley's Theorem 3.7: Lagrange's Theorem 3.8: Orbits 3.9: Orbit/Stabilizer Theorem 3.10: Burnside's Theorem 3.11: K-Colourings 3.12: Cycle Index and Enumeration 4: Groups, Homomorphisms and Subgroups 4.1: Homomorphisms 4.2: The Isomorphism Theorems 4.3: Direct Products 4.4: Finite Abelian Groups 4.5: Conjugacy and the Class Equation 4.6: The Sylow Theorems 1 and 2 4.7: Sylow's Third Theorem 4.8: Solvable Groups 4.9: Nilpotent Groups 5: Rings and Polynomials 5.1: Homomorphisms and Ideals 5.2: Polynomial Rings 5.3: Divison Algorithm in F[x1, x2, ..., xn]; Single Divisor 5.4: Multiple Divisors; Groebner Bases 5.5: Ideals and Affine Varieties 5.6: Complex Numbers 5.7: Decomposition of Affine Varieties 5.8: Cubic Equations in One Variable 5.9: Parameters 5.10: Intersection Multiplicities 5.11: Singular and Nonsingular Points 6: Elliptic Curves 6.1: Elliptic Curves 6.2: Homogeneous Polynomials 6.3: Projective Space 6.4: Intersection of Lines and Curves 6.5: Defining Curves by Points 6.6: Classification of Conics 6.7: Reducible Conics and Cubics 6.8: The Nine Point Theorem 6.9: Groups on Elliptic Curves 6.10: The Arithmetic on an Elliptic Curve 6.11: Results Concerning the Structure of Groups on Elliptic Curves 7: Further Topics Relating to Elliptic Curves 418 7.1: Elliptic Curve Cryptosystems 7.2: Fermat's Last Theorem 7.3: Elliptic Curve Factoring Algorithm 7.4: Singular Curves of Form y2=x3+ax+b 7.5: Birational Equivalence 7.6: The Genus of a Curve 7.7: Pell's Equation Bibliography

About the Author :
Norman Reilly has been an educator for almost 50 years, starting in high school in Scotland and proceeding to Universities in Scotland, America and Canada. He has published one book and over 80 research papers in algebra. He was born in Glasgow, Scotland. Books by the same author: Completely Regular Semigroups |a Mario Petrich and Norman Reilly

Review :
"An excellent source for the advanced undergraduate seeking to make the transition to higher mathematics. Future computer scientists and number theorists interested in applications will find this book a good foundation text." -- MAA Reviews "Ideal for a course in abstract algebra, providing students with a sound background in algebra, while teaching them many current-day uses of algebra in the real world. The reviewer was particularly impressed with the flow and depth provided for each application and found the discussions fascinating."--Libertas Mathematica "The whole book would be suitable as a text for a year-long course in algebra, or the first chapter by itself for a short course in number theory, or the first three chapters for a semester-long course in algebra. It is noteworthy for its inclusion of extensive historical material on breaking the Enigma code in World War II and the theorems used to accomplish this." -- SIAM Review