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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:
Table of Contents
Chapter 1. New Numbers 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
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CHOICE – Outstanding Academic Title – Award Winner
CHOICE – 2018 Outstanding Academic Title – Award Winner
Shingo Research and Professional Publication Award Winner
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