Elements of algebra : geometry, numbers, equations
著者
書誌事項
Elements of algebra : geometry, numbers, equations
（Undergraduate texts in mathematics）
Springer-Verlag, c1994
- : us
- : gw
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注記
Includes bibliographical references (p. [162]-169) and index
内容説明・目次
- 巻冊次
-
: us ISBN 9780387942902
内容説明
Algebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart.
目次
- Preface
- 1. Algebra and Geometry
- 2. The Rational Numbers
- 3. Numbers In General
- 4. Polynomials
- 5. Fields
- 6. Isomorphisms
- 7. Groups
- 8. Galois Theory of Unsolvability
- 9. Galois Theory of Solvability
- References
- Index
- 巻冊次
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: gw ISBN 9783540942900
内容説明
This text is a concise, self-contained introduction to abstract algebra which stresses its unifying role in geometry and number theory. There is a strong emphasis on historical motivation - both to trace abstract concepts to their concrete roots, but also to show the power of new ideas to solve old problems. This approach shows algebra as an integral part of mathematics and makes this text more informative to both beginners and experts than others. Classical results of geometry and number theory (such as straightedge-and-compass construction and its relation to Fermat primes) are used to motivate and illustrate algebraic techniques, and classical algebra itself (solutions of cubic and quartic equations) is used to motivate the problem of solvability by radicals and its solution via Galois theory. Modern methods are used whenever they are clearer or more efficient, but technical machinery is introduced only when needed.
目次
Preface.- Algebra and Geometry.- The Rational Numbers.- Numbers in General.- Polynomials.- Fields.- Isomorphisms.- Groups.- Galois Theory of Unsolvability.- Galois Theory of Solvability.- References. Index.
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