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| Galois Theory (Graduate Texts in Mathematics)
Hardcover
Author: Harold M. Edwards
Publisher: Springer
Release Date: 1997-11-26
ISBN-10: 038790980X
ISBN-13: 9780387909806
List Price: $69.95
Average Customer Rating:
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This book is an introduction to Galois theory along the lines of Galois' "Memoir on the Conditions for Solvability of Equations by Radicals". Some antecedents of Galois theory in the works of Gauss, Lagrange, Vandemonde, Newton, and even the ancient Babylonians, are explained in order to put Galois' main ideas in their historical setting. The modern formulation of the theory is also explained. The book contains many exercises - with answers - and an English translation of Galois' memoir.
Breath the air of Galois
| Customer Rating: | This is not an excellent exposition of Galois theory. It is, however, a book well worth reading for the single reason that it sticks to Galois, including a full translation of Galois' 1831 memoir (13 pages).
The immediate goal for Galois was to understand solvability by radicals, in particular of the general n:th degree equation. But to understand Galois we must first study what was done before him (§§1-27). Lagrange is the most important predecessor. He presented a unified approach to the solvable cases (n<5) in terms of "resolvents", which he hoped would point forward to the n>4 cases. This resolvent business is historical ballast, we would say today, but it is from this tradition that Galois departs. In particular, he arrives at the "Galois group" in terms of resolvents. With the Galois group in place, things flow more smoothly. Essentially as in the modern theory, Galois shows that if an equation is solvable by radicals then its Galois group is "solvable". All this is §§28-48. Edwards the constructivist now inserts a bunch of Kronecker material on the existence of roots (§§49-61). Then it's back to Galois (§§62-71) to see how he puts his theory to use. Galois doesn't even bother to spell out that the unsolvability of the general equation of degree n>4 follows since its Galois group S_n is not solvable; instead he finds a curious criterion for solvability which is involves no group theory (even Edwards calls this result "rather strange"). |
Best treatment for the "mechanics" of Galois Theory
| Customer Rating: | | This is the only book I have seen that "mechanically dissects" galois theory. Other books would give a succession of "theorem-proofs" that eventually proved the Galois solvability theorem but when I was finished, I still could not intuitively understand how the solvability of Galois groups to corresponds to solvability by radicals. Instead of plodding thru theorem-proof without insight or motivation, this book actually works thru the relation between the algebra of polynomial solutions and the structure of their groups. Ultimately it shows the structure that an equation must have to be solvable (equivalently to have a solvable Galois group). Unlike the other reader who did not get alot out of Galois' original writing, it opened a window of understanding for me. Make no mistake, understanding the material here took considerable effort to work out the math on paper to follow the examples and the proofs. Perhaps the book could be improved by adding more detail in places, but then, this is a graduate-level text. This is the second book I have read from Harold Edwards. I found I learn an awful lot of the low-level details of a subject that I can't find anywhere else. I believe he is one of the best authors in mathematics today. I would happily collect all his works. |
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