1st Edition
Numbers and Symmetry An Introduction to Algebra
This textbook presents modern algebra from the ground up using numbers and symmetry. The idea of a ring and of a field are introduced in the context of concrete number systems. Groups arise from considering transformations of simple geometric objects. The analysis of symmetry provides the student with a visual introduction to the central algebraic notion of isomorphism.
Designed for a typical one-semester undergraduate course in modern algebra, it provides a gentle introduction to the subject by allowing students to see the ideas at work in accessible examples, rather than plunging them immediately into a sea of formalism. The student is involved at once with interesting algebraic structures, such as the Gaussian integers and the various rings of integers modulo n, and is encouraged to take the time to explore and become familiar with those structures.
In terms of classical algebraic structures, the text divides roughly into three parts:
A Planeful of Integers, Z[i]
Circular Numbers, Zn
More Integers on the Number Line, Z [v2]
Notes
Chapter 2. The Division Algorithm
Rational Integers
Norms
Gaussian Numbers
Q (v2)
Polynomials
Notes
Chapter 3. The Euclidean Algorithm
Bézout's Equation
Relatively Prime Numbers
Gaussian Integers
Notes
Chapter 4. Units
Elementary Properties
Bézout's Equations
Wilson's Theorem
Orders of Elements: Fermat and Euler
Quadratic Residues
Z [v2]
Notes
Chapter 5. Primes
Prime Numbers
Gaussian Primes
Z [v2]
Unique Factorization into Primes
Zn
Notes
Chapter 6. Symmetries
Symmetries of Figures in the Plane
Groups
The Cycle Structure of a Permutation
Cyclic Groups
The Alternating Groups
Notes
Chapter 7. Matrices
Symmetries and Coordinates
Two-by-Two Matrices
The Ring of Matrices M2(R)
Units
Complex Numbers and Quaternions
Notes
Chapter 8. Groups
Abstract Groups
Subgroups and Cosets
Isomorphism
The Group of Units of a Finite Field
Products of Groups
The Euclidean Groups E (1), E (2), and E (3)
Notes
Chapter 9. Wallpaper Patterns
One-Dimensional Patterns
Plane Lattices
Frieze Patterns
Space Groups
The 17 Plane Groups
Notes
Chapter 10. Fields
Polynomials Over a Field
Kronecker's Construction of Simple Field Extensions
Finite Fields
Notes
Chapter 11. Linear Algebra
Vector Spaces
Matrices
Row Space and Echelon Form
Inverses and Elementary Matrices
Determinants
Notes
Chapter 12. Error-Correcting Codes
Coding for Redundancy
Linear Codes
Parity-Check Matrices
Cyclic Codes
BCH Codes
CDs
Notes
Chapter 13. Appendix: Induction
Formulating the n-th Statement
The Domino Theory: Iteration
Formulating the Induction Statement
Squares
Templates
Recursion
Notes
Chapter 14. Appendix: The Usual Rules
Rings
Notes
Index
Biography
Department of Mathematical Sciences Florida Atlantic University Boca Raton, Florida. Department of Mathematical Sciences Florida Atlantic University Boca Raton, Florida.
