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Introduction to Applied Algebraic Systems

Norman R. Reilly
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
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Introduction to Applied Algebraic Systems is book of breadth that can support two semesters of lecture and study. This could be at the upper-level undergraduate or first-year graduate level. The title may suggest to some that applications are germane to the content. However, after a quick overview of UPC codes to exemplify applied modular arithmetic in Chapter 1, tightly coupled applications do not return until the final seventh chapter, on elliptic curves.

The core five chapters are a rigorous introduction to number theory, encompassing rings, fields, polynomial theory, groups, and algebraic geometry. This all leads to and supports two chapters on elliptic curves. Thus the book delivers a number theory foundation from modular arithmetic through basic fields on to algebraically closed fields, affine varieties and ideals. Each section concludes with about a dozen exercises well-motivated by the previous proofs and comments. The independent reader will find no solutions for these, but the frequent introduction of exercises after new ideas makes this manageable.

The final chapter, the second one on elliptic curves, is entitled “Further Topics Related to Elliptic Curves.” The reader that has made it this far will be prepared to understand and receive a glimpse of the application of these curves to cryptography, their use in prime factoring, and the role they played in Wiles’ proof of Fermat’s Last Theorem. The final topic introduces the idea of the genus of a curve.

Future computer scientists and number theorists interested in applications will find this book a good foundation text. Also, this is an excellent source for the advanced undergraduate seeking to make the transition to higher mathematics.

Tom Schulte first met elliptic curves in graduate study at Oakland University. He lives, works, and teaches in Michigan.

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 Eulernulls 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 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 Cayleynulls Theorem
3.7 Lagrangenulls Theorem
3.8 Orbits
3.9 Orbit/Stabilizer Theorem
3.10 The Cauchy-Frobenius Theorem
3.11 K-Colorings
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 Sylownulls Third Theorem
4.8 Solvable Groups
4.9 Nilpotent Groups
4.10 The Enigma Encryption Machine

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 Decomposition of Affine Varieties
5.7 Cubic Equations in One Variable
5.8 Parameters
5.9 Intersection Multiplicities
5.10 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 Related to Elliptic Curves
7.1 Elliptic Curve Cryptosystems
7.2 Fermatnulls 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 Pellnulls Equation