Norman R. Reilly

Introduction to Applied Algebraic Systems

• Gebundenes Buch

Produktbeschreibung

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.

Produktdetails

• Verlag: OXFORD UNIV PR
• Seitenzahl: 524
• Erscheinungstermin: 1. Dezember 2009
• Englisch
• Abmessung: 240mm x 161mm x 33mm
• Gewicht: 945g
• ISBN-13: 9780195367874
• ISBN-10: 0195367871
• Artikelnr.: 26100456

Autorenporträt

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

Inhaltsangabe

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: Multiplicative 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
3.12Cycle 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: Division 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
5.105.11: 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