Galois' Theory of Algebraic Equations | 
 enlarge | Author: Jean-pierre Tignol
Publisher: World Scientific Publishing Company
Category: Book
List Price: $34.00
Buy New: $30.60
You Save: $3.40 (10%)
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Rating:  2 reviews
Sales Rank: 695851
Media: Paperback
Pages: 333
Number Of Items: 1
Shipping Weight (lbs): 1.1
Dimensions (in): 8.4 x 6 x 0.8
ISBN: 9810245416
Dewey Decimal Number: 512.3
EAN: 9789810245412
Publication Date: July 2001
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Product Description
Galois' Theory of Algebraic Equations gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The main emphasis is placed on equations of at least the third degree, i.e. on the developments during the period from the sixteenth to the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as "group" and "field". A brief discussion on the fundamental theorems of modern Galois theory is included. Complete proofs of the quoted results are provided, but the material has been organized in such a way that the most technical details can be skipped by readers who are interested primarily in a broad survey of the theory. This book will appeal to both undergraduate and graduate students in mathematics and the history of science, and also to teachers and mathematicians who wish to obtain a historical perspective of the field. The text has been designed to be self-contained, but some familiarity with basic mathematical structures and with some elementary notions of linear algebra is desirable for a good understanding of the technical discussions in the later chapters.
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Another interesting, historical Galois Theory book June 2, 2006
Viktor Blasjo
12 out of 14 found this review helpful
After some one hundred pages of semi-historical digressions on various scattered topics in classical algebra, we get to the main topic: the question of solvability by radicals of polynomial equations. The known solutions of equations up to degree 4 can be interpreted as relying on certain solvable auxiliary equations, the roots of which can be used to express the roots of the original equation. Lagrange then introduced the would-be analogous auxiliary equations for higher degree cases. Their roots, "Lagrange resolvents", can express the roots of the original equation, but their appearance suggests that they will not be solvable. However, proving this sort of thing seems to call for "a kind of calculus of combinations" (Lagrange; p. 146), i.e. permutation group theory, beyond his reach. The subsequent development followed the lines indicated by Lagrange to some extent; Gauss's proof that cyclotomic equations are solvable by radicals is based on solving iterated auxiliary equations, thus providing "remarkable examples of the step-by-step solution of equation as envisioned by Lagrange" (p. 185), and the Ruffini-Abel unsolvability proofs did indeed involve a little bit of permutation groups. But the paramount vindication and perfection came with Galois. Galois has his own "resolvents"--given an equation, a Galois resolvent is a calculable expression that can rationally express all the roots of the equation. Now, if one substitutes into these rational expressions another root of the minimal polynomial of the resolvent, then one still gets the roots, but they are permuted. All such permutations form a group--the Galois group--which is the key to solvability. Namely, solvability by radicals of the equation, i.e. solvability by +,-,*,/ and p:th roots, is precisely mirrored by the "solvability" of the corresponding Galois group, i.e. the decomposition of this group into a chain of normal subgroups of index p.
an idea of how mathematics is made January 22, 2002
15 out of 19 found this review helpful
I have tried for several months to find a book wich could give me some inside of "GALOIS Theory", an idea of how people came to such abstract considerations.
I think I've found it!
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