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I always feel that it is possible to teach abstract algebra in a simple way that follows the historical developments. An example is it is totally reasonable for me to start learning basic Galois theory after one is comfortable with linear algebra. By learning Galois theory in this way, the needs of group theory and ring (or say polynomial) theory naturally arise, and most of the materials in Dummit & Foote can be also covered.

The only obstruction I have noticed to teaching in this way is; the standard textbooks were not written in this fashion, and it is too hard to rewrite one while you know probably no one except you will use your note. However, I still believe such lecture-note/references exist, and would like to ask if you know any of such. Thank you so much.

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    $\begingroup$ The reason textbooks develop ring and group theory before talking about Galois theory is because it's an absolute clusterf*ck if you don't. For example, try to read Abel's original paper on the unsolvability of the quintic. $\endgroup$ – D_S May 16 '18 at 21:34
  • $\begingroup$ I agree that we should not follow entirely the history, but it is still possible to learn Galois theory first. $\endgroup$ – Student May 16 '18 at 21:36
  • $\begingroup$ @D_S An even better argument is probably provided by one of Ruffini's infamously incomprehensible 500-page efforts! $\endgroup$ – Chappers May 16 '18 at 21:42
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John Stillwell had a wonderful little article "Galois Theory for Beginners" that appeared in the American Mathematical Monthly in January of 1994 (pp. 22-27). You might be interested in it. I was always struck by the following sentence from the introduction:"I read the books of Edwards, Tignol, Artin, ... , and Lang, taught a course in Galois theory, and then discarded 90% of what I learned." Happy reading!

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Ian Stewart's book begins with the historical development, and classical algebra involved in studying roots of polynomials. I can't recall if you need the definitions of groups etc., but you definitely do not need much of the theory. Also the first half of the book is with complex polynomials, so no ring/field theory is really required at all.

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Tignol's Galois' Theory of Algebraic Equations is an historically-informed build-up to and introduction to Galois' original paper. (Its history varies between okay and just plain wrong, but the mathematics is soundly developed.) It's probably the only place you'll find a textbook-ish treatment of Lagrange's theory, for example, as well as several different ways of solving the cubic and quartic (Cardano, Descartes, Viète, Tschirnhaus, ...), complete with the dead-ends for the quintic.

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Harold Edwards' Galois Theory aims at following Galois' work and historical advances as closely as possible.

You can find a comprehensive review by Peter M. Neumann: see on JSTOR or on the MAA website. He won the Lester R. Ford award from the MAA for this review.

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