Galois Theory (Graduate Texts in Mathematics, 101) by Harold M. Edwards (PDF)

Description

This is an introduction to Galois Theory along the lines of Galois’s Memoir on the Conditions for Solvability of Equations by Radicals. It puts Galois’s ideas into historical perspective by tracing their antecedents in the works of Gauss, Lagrange, Newton, and even the ancient Babylonians. It also explains the modern formulation of the theory. It includes many exercises, with their answers, and an English translation of Galois’s memoir.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Very well written.

⭐I thoroughly enjoyed this book. It’s a bit weird that other Edwards books have a live link for his name which leads to eight books, none of which are this one. Very strange. Anyway, this is a really nice read that made me understand how a 20 year old kid managed to prove Abel’s theorem. When I learned Galois Theory in school it seems ridiculous that a kid could have done that, especially one who hadn’t learned any math until he was 15 and then had to do part of the work in prison. It’s still one of the most herculean feats in all of math history, but Edwards makes you understand how he approached it in a much more down-to-earth way than is now presented in textbooks. For example he didn’t really need the concept of a solvable series of normal subgroup, he just worked with things with prime index and never really needed to get abstract like that. In the end he walks you through Galois actual notes. It’s a real beautiful experience. Edwards books are unique, his style is unique and quite refreshing.

⭐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.

⭐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”).

⭐Interessante dal punto di vista storico ma sconsigliato a chi si approccia alla materia per la prima volta in quanto estremamente conciso.

⭐

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