The famous problems of squaring the circle, doubling the cube, and trisecting the angle have captured the imagination of both professional and amateur mathematician for over two thousand years. These problems, however, have not yielded to purely geometrical methods. It was only the development of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. This text aims to develop the abstract algebra.
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6 of 6 people found the following review helpful:
5.0 out of 5 stars
The easiest introduction to field extensions I've read,
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This review is from: Abstract Algebra and Famous Impossibilities (Universitext) (Paperback)
This is the easiest-to-read introduction to abstract algebra and, in particular, field extensions, that I have ever seen. It is easier to follow than Artin's famous little book, and has easy but useful problems to allow you to check your understanding.This material is developed in order to explain the questions of how to prove that certain famously unsolved problems of geometry (a general Euclidean method for trisecting an arbitrary angle, constructing a square with the area of a given circle) are actually unsolvable. It also addresses the question of the transcendence of Euler's number (e). The first problem is handled by understanding the abstract algebra of geometrical constructions; the second goes further into analysis as well; and I haven't finished reading the discussion on the third problem. Admittedly, the learning curve rises rather rapidly on the last two problems! However, the book is worth reading even for the first problem alone, and as an introduction to abstract algebra.
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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57
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- Notes on Fermat's Last Theorem (Canadian Mathematical Society Series of Monographs and Advanced Texts) by Alf J. van der Poorten on page 27, and page 39
- A Short Course on Operator Semigroups (Universitext) by Klaus-Jochen Engel in Back Matter
- Heights of Polynomials and Entropy in Algebraic Dynamics (Universitext) by Graham Everest in Front Matter
- Introduction to Game Theory (Universitext) by Peter Morris in Front Matter
- Lectures on Analysis on Metric Spaces (Universitext) by Juha Heinonen in Front Matter
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