Abstract algebra

By: Ronald SolomonMaterial type: TextTextPublication details: Rhiode Island: American Mathematical Society, [c2003]Description: 227 pISBN: 9780821852101Subject(s): MathematicsLOC classification: QA162
Contents:
Introduction Chapter 0. Background 1. What Is Congruence? 2. Some Two-Dimensional Geometry 3. Symmetry 4. The Root of It All 5. The Renaissance of Algebra 6. Complex Numbers 7. Symmetric Polynomials and The Fundamental Theorem of Algebra 8. Permutations and Lagrange’s Theorem 9. Orbits and Cauchy’s Formula 9A. Hamilton’s Quaternions (Optional) 10. Back to Euclid 11. Euclid’s Lemma for Polynomials 12. Fermat and the Rebirth of Number Theory 13. Lagrange’s Theorem Revisited 14. Rings and Squares 14A. More Rings and More Squares 15. Fermat’s Last Theorem (for Polynomials) 15A. Still more Fermat’s Last Theorem (Optional) 16. Constmctible Polygons and the Method of Mr. Gauss 17. Cyclotomic Fields and Linear Algebra 18. A Lagrange Theorem for Fields and Nonconstructibility 19. Galois Fields and the Fundamental Theorem of Algebra Revisited 20. Galois’ Theory of Equations 21. The Galois Correspondence 22. Constructible Numbers and Solvable Equations
Summary: This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter. --- summary provided by publisher
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Item type Current library Collection Shelving location Call number Status Notes Date due Barcode Item holds
Book Book ICTS
Mathematic Rack No 4 QA162 (Browse shelf (Opens below)) Available Billno:IN 003 582; Billdate: 2018-01-11 00867
Total holds: 0

Introduction
Chapter 0. Background
1. What Is Congruence?
2. Some Two-Dimensional Geometry
3. Symmetry
4. The Root of It All
5. The Renaissance of Algebra
6. Complex Numbers
7. Symmetric Polynomials and The Fundamental Theorem of Algebra
8. Permutations and Lagrange’s Theorem
9. Orbits and Cauchy’s Formula
9A. Hamilton’s Quaternions (Optional)
10. Back to Euclid
11. Euclid’s Lemma for Polynomials
12. Fermat and the Rebirth of Number Theory
13. Lagrange’s Theorem Revisited
14. Rings and Squares
14A. More Rings and More Squares
15. Fermat’s Last Theorem (for Polynomials)
15A. Still more Fermat’s Last Theorem (Optional)
16. Constmctible Polygons and the Method of Mr. Gauss
17. Cyclotomic Fields and Linear Algebra
18. A Lagrange Theorem for Fields and Nonconstructibility
19. Galois Fields and the Fundamental Theorem of Algebra Revisited
20. Galois’ Theory of Equations
21. The Galois Correspondence
22. Constructible Numbers and Solvable Equations

This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter. --- summary provided by publisher

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