Abstract Algebra

by Solomon, Ronald

• ISBN13: 9780534399962
• ISBN10: 0534399967
• Edition: 1st
• Format: Paperback
• Copyright: 2002-12-16
• Publisher: Brooks Cole

Summary

In the same way the subject evolved historically, Ronald Solomon lets the abstract concepts emerge gradually from less abstract problems about geometry, polynomials, numbers, and more. Solomon also strongly emphasizes the connections between algebra and other areas of mathematics -- analysis (the infinitesimal calculus) and geometry. Readers will see that the various areas of mathematics are not hermetically sealed off from each other and that most of the truly important achievements in mathematics have been the product of a fruitful interaction of different areas. Using this book, readers will gain a true understanding of the subject while being creative mathematically as opposed to simply imitating template problems.

Table of Contents

Background	7	(6)
Section One Geometry	13	(32)
What Is Congruence?	14	(8)
Some Two-Dimensional Geometry	22	(10)
Symmetry	32	(13)
Section Two Polynomials	45	(56)
The Root of It All	46	(3)
The Renaissance of Algebra	49	(10)
Complex Numbers	59	(9)
Symmetric Polynomials and The Fundamental Theorem of Algebra	68	(10)
Permutations and Lagrange's Theorem	78	(8)
Orbits and Cauchy's Formula	86	(9)
Hamilton's Quaternions (Optional)	95	(6)
Section Three Numbers	101	(68)
Back to Euclid	102	(11)
Euclid's Lemma for Polynomials	113	(9)
Fermat and the Rebirth of Number Theory	122	(14)
Lagrange's Theorem Revisited	136	(6)
Rings and Squares	142	(7)
More Rings and More Squares	149	(8)
Fermat's Last Theorem (for Polynomials)	157	(8)
Still more Fermat's Last Theorem (Optional)	165	(4)
Section Four The Grand Synthesis	169	(54)
Constructible Polygons and the Method of Mr. Gauss	170	(7)
Cyclotomic Fields and Linear Algebra	177	(14)
A Lagrange Theorem for Fields and Nonconstructibility	191	(5)
Galois Fields and the Fundamental Theorem of Algebra Revisited	196	(11)
Galois' Theory of Equations	207	(5)
The Galois Correspondence	212	(5)
Constructible Numbers and Solvable Equations	217	(6)
Index	223