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| Galois Theory for Beginners: A Historical Perspective (Student Mathematical Library) (Student Matehmatical Library)
Paperback
Author: Jorg Bewersdorff
Publisher: American Mathematical Society
Release Date: 2006-09-05
ISBN-10: 0821838172
ISBN-13: 9780821838174
List Price: $35.00
Average Customer Rating:
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Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book, Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of particular investigations. Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed. Results in Galois theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular $n$-gons are also presented. This book is suitable for undergraduates and beginning graduate students.
A highly readable introduction to Galois groups
| Customer Rating: | This is a wonderfully written, highly accessible introduction to polynomial equations (quadratics, cubics, etc.) and their solutions by radicals, leading step by step to an introduction to Galois theory.
Galois theory is presented only towards the end of the book. Readers already familiar with the solutions of quadratic equations, depressed cubics, cubics, and quartics will find the first half of the book somewhat redundant. But it is nevertheless very pleasant to read, with succinct notes on the historical background, and (mostly) self-contained short sections.
It reads very well all the way to the end. It gets a little harder when Galois theory is introduced. But that's perhaps to be expected. I can't say that I master the subject, but certain things (about polynomial equations) are a great deal clearer for me now.
I do have one reservation (but I did not knock off a star for that): the editing (of this English translation of the German original) is quite poor: there is a typo just about every other page. I am very sensitive to typos, and most readers probably won't (nor should they) care -- but there are some typos in the math here and there, and that's plain unacceptable. |
A solvable polynomial has a solvable Galois group. A constructible point in a plane is similar to a solvable solution.
| Customer Rating: | The main focus of Jorg Bewersdorff's "Galois Theory for Beginners: An Historical Perspective" is "...[polynomials] and their solutions..." Chapters 1 through 3 are about classical methods (formulas) for solutions on cubic and bi-quadratic polynomials. Chapter 4 and 5 are "systematic investigation of the...solution formulas..." Chapter 6's primary topic is "...[polynomials] that can be broken down into...lower degree." Chapter 7 focuses on the construction of regular polygons with straightedge and compass. Chapter 8 is about "...finding a general solution formula for [the] fifth-degree [polynomials]..." Chapter 9 and 10 focus on Galois theory: "...the limits of solvability of [polynomials] in radicals."
I began my pursuit "...to understand why a general...[polynomial] of the 5th degree should have no solution in radical..." decades ago, as if Jorg Bewersdorff. His book is the best I have ever read on Galois theories. For instances:
1. Theorem 10.18 An [polynomial] is solvable in radicals, that is all of its solutions can be expressed in terms of nested roots whose radicands can be expressed in terms of the coefficients using the four basic operations, if and only if its Galois group is solvable...
2. Definition 9.2 For a [polynomial] without multiple solutions whose coefficients lie in a field K, the Galois group (over the field K) is the set of all permutations s in the symmetric group Sn that permute the indices 1,...,n of the solutions x1,...,xn in such a way that for every polynomial h(X1,...,Xn) with coefficients in K and h(x1,x2,...,xn) = 0, one has h(x(s(1)),...,x(s(n))) = 0 ...
3. Definition 10.17 A finite group G is called solvable if there is a chains of groups {id}=G0 ( G1 ( G2 ( ... ( G(k-1) ( G(k) = G for which the subgroup G(j) is a normal subgroup of the next group in the chain G(j+1), such that the quotient group G(j+1)/G(j) is cyclic of prime order n...
4. A point in a plane is constructible if and only if its "...coordinates can be expressed in rational numbers and nested square roots using the four basic arithmetic operations (+, - , * , /).
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