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I know that there have been written similar posts, and I used them as a source for my question.

I ' m looking for a book for Galois Theory (Construction of fields. Algebraic extensions - Classical Greek problems: constructions with ruler and compass. Galois extensions - Applications: solvability of algebraic equations - The fundamental theorem of Algebra - Roots of unity - Finite fields), which has the following characteristics:

  • Logical order in the presentation of the theorems, definitions and generally of all concepts.
  • Thorough analysis of each proof, example etc.
  • Many examples and good exercises to solve.
  • Also, to be suitable for self-study and for the first touch in the subject.

I should notice that I don't like Stewart's and Rotman's book.

What's your opinion for 1) Galois Theory by Bakers, 2) Galois Theory by Roman, 3) Fields and Galois Theory by Howie, 4) Galois Theory by Jean-Pierre Escofier. And do you believe that it is better to read from a general Abstract Algebra book, such that Fraleigh's/ Dummit's and Foote's/Gallian's?

Thank you in advance.

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  • $\begingroup$ While this is a specific question, the second part is obviously highly opinion-based. You should remove it. $\endgroup$ – k.stm Jul 18 '17 at 7:30
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I recommend Galois' Theory of Algebraic Equations, by Jean-Pierre Tignol (2nd edition, World Scientific, 2016).

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    $\begingroup$ Tignol's book is especially notable for its historical information. (I haven't seen the 2nd edition, however.) For a lot of specific examples and calculations, see Classical Galois Theory, With Examples by Lisl Gaal. $\endgroup$ – Dave L. Renfro Jul 10 '17 at 14:42
  • $\begingroup$ @DaveL.Renfro I strongly agree with you. $\endgroup$ – José Carlos Santos Jul 10 '17 at 14:43
  • $\begingroup$ @JoséCarlosSantos I found Tignol's book great, but it's not what I was looking for. It's more historic and less an introduction to classical Galois Theory. $\endgroup$ – Chris Jul 10 '17 at 17:48
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    $\begingroup$ Chris, you might want to look at the treatments in Fraleigh's A First Course in Abstract Algebra and Herstein's Topics in Algebra. Fraleigh's treatment is especially elementary and excellent for self-study (I read much of the 1st edition of his book on my own), but it might be too limited in scope for your needs. Herstein's treatment is a little more advanced, but still very well written, which I can personally testify to, having taken a 2-semester algebra sequence using his text. $\endgroup$ – Dave L. Renfro Jul 11 '17 at 14:43
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    $\begingroup$ Also, you may want to look at Hadlock's Field Theory and Its Classical Problems, and perhaps scan through the bibliography of this manuscript (also located here). The items in this bibliography were not chosen for someone interested in Galois theory, but some of them might still be of use to you. $\endgroup$ – Dave L. Renfro Jul 11 '17 at 14:51
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You clearly know which books are available on the subject. I would like to comment on your request. "Also, to be suitable for self-study and for the first touch in the subject."

The lecture notes of Christopher Cooper on Galois Theory ( Math338 ) are just that: suitable for self-study, and for a first touch in the subject.

https://rutherglen.science.mq.edu.au/ccooper/Galois/

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    $\begingroup$ Thank you for your post. These notes are very useful (and with answers in exercises)! $\endgroup$ – Chris Jul 18 '17 at 8:38
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I’ve learned Galois Theory from Siegfried Bosch’s Book Algebra. It’s German, though. To my knowledge, it has been translated to English, but I do not know how available English versions are.

Anyway, it pretty much fits the bill: It is very clean, leaves absolutely nothing out (not even set-theoretical considerations when discussion transcendence degree), it is well-structured and yet light-to-read. And it covers, of course, everything you have listed.

It also has additional (starred) sections on more advanced topics. I most of all value his introductions to each chapter which motivate the following material really well and give a good overview of what to expect (which makes it really suitable for self-study). The introductory chapter also gives some historical context.

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  • $\begingroup$ Thank you for your post. This book looks pretty good but I have searched for it and I couldn't find it in an English version! $\endgroup$ – Chris Jul 18 '17 at 8:36

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