Introduction to Applied Algebraic Systems by Norman R Reilly

Introduction to Applied Algebraic Systems

byNorman R Reilly

Hardcover | December 4, 2009

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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 forexploring 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 numbertheory 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 algebraicsystems 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 sciencestudents 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.

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. Born in Glasgow, Scotland.
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Details & Specs

Title:Introduction to Applied Algebraic SystemsFormat:HardcoverDimensions:464 pages, 9.25 × 6.13 × 0.98 inPublished:December 4, 2009Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0195367871

ISBN - 13:9780195367874

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Extra Content

Table of Contents

1. Modular Arithmetic1.1 Sets, functions, numbers1.2 Induction1.3 Divisibility1.4 Prime Numbers1.5 Relations and Partitions1.6 Modular Arithmetic1.7 Equations in Zn1.8 Bar codes1.9 The Chinese Remainder Theorem1.10 Euler's '-function1.11 Theorems of Euler and Fermat1.12 Public Key Cryptosystems2. Rings and Fields2.1 Basic Properties2.2 Subrings and Subfields2.3 Review of Vector Spaces2.4 Polynomials2.5 Polynomial Evaluation and Interpolation2.6 Irreducible Polynomials2.7 Construction of Finite Fields2.8 Extension Fields2.9 Multiplicative Structure of Finite Fields2.10 Primitive Elements2.11 Subfield Structure of Finite Fields2.12 Minimal Polynomials2.13 Isomorphisms Between Fields2.14 Error Correcting Codes3. Groups and Permutations3.1 Basic Properties3.2 Subgroups3.3 Permutation Groups3.4 Matrix Groups3.5 Even and Odd Permutations3.6 Cayley's Theorem3.7 Lagrange's Theorem3.8 Orbits3.9 Orbit/Stabilizer Theorem3.10 Burnside's Theorem3.11 K-Colourings3.12 Cycle Index and Enumeration4. Groups; Homomorphisms and Subgroups4.1 Homomorphisms4.2 The Isomorphism Theorems4.3 Direct Products4.4 Finite Abelian Groups4.5 Conjugacy and the Class Equation4.6 The Sylow Theorems 1 and 24.7 Sylow's Third Theorem4.8 Solvable Groups4.9 Nilpotent Groups5. Rings and Polynomials5.1 Homomorphisms and Ideals5.2 Polynomial Rings5.3 Division Algorithm in F[x1, x2, . . . , xn]; Single Divisor5.4 Multiple Divisors; Groebner Bases5.5 Ideals and Affine Varieties5.6 Complex Numbers5.7 Decomposition of Affine Varieties5.8 Cubic Equations in One Variable5.9 Parameters5.10 Intersection Multiplicities5.11 Singular and Nonsingular Points6. Elliptic Curves6.1 Elliptic Curves6.2 Homogeneous Polynomials6.3 Projective Space6.4 Intersection of Lines and Curves6.5 Defining Curves by Points6.6 Classification of Conics6.7 Reducible Conics and Cubics6.8 The Nine Point Theorem6.9 Groups on Elliptic Curves6.10 The Arithmetic on an Elliptic Curve6.11 Results Concerning the Structure of Groups on Elliptic Curves7. Further Topics Relating to Elliptic Curves 4187.1 Elliptic Curve Cryptosystems7.2 Fermat's Last Theorem7.3 Elliptic Curve Factoring Algorithm7.4 Singular Curves of Form y2 = x3 + ax + b7.5 Birational Equivalence7.6 The Genus of a Curve7.7 Pell's EquationBibliography