Abstract Algebra 1st Edition by Ronald Solomon ISBN 0821852108 978-0821852101

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Product details:

ISBN 10: 0821852108 

ISBN 13: 978-0821852101

Author: Ronald Solomon

 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.

Table of contents:

Section One – Geometry
What Is Congruence?
Some Two-Dimensional Geometry
Symmetry
Section Two – Polynomials
The Root of It All
The Renaissance of Algebra
Complex Numbers
Symmetric Polynomials and The Fundamental Theorem of Algebra
Permutations and Lagrange’s Theorem
Orbits and Cauchy’s Formula
9A. Hamilton’s Quaternions (Optional)
Section Three – Numbers
Back to Euclid
Euclid’s Lemma for Polynomials
Fermat and the Rebirth of Number Theory
Lagrange’s Theorem Revisited
Rings and Squares
14A. More Rings and More Squares
Fermat’s Last Theorem (for Polynomials)
15A. Still more Fermat’s Last Theorem (Optional)
Section Four – The Grand Synthesis
Constructible Polygons and the Method of Mr. Gauss
Cyclotomic Fields and Linear Algebra
A Lagrange Theorem for Fields and Nonconstructibility
Galois Fields and the Fundamental Theorem of Algebra Revisited
Galois’ Theory of Equations
The Galois Correspondence
Constructible Numbers and Solvable Equations

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