Galois Theory books (in association with Abstract Algebra books)

Question

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.

While this is a specific question, the second part is obviously highly opinion-based. You should remove it. — k.stm, Jul 18 '17 at 07:30
Thanks for your comment. It might be opinion based, however I think I will be helpful for many people. — Chris, May 21 '21 at 17:36

score 3 · Answer 1 · answered Jul 10 '17 at 13:44

3

I recommend Galois' Theory of Algebraic Equations, by Jean-Pierre Tignol (2nd edition, World Scientific, 2016).

answered Jul 10 '17 at 13:44

José Carlos Santos

440,053

2

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. – Dave L. Renfro Jul 10 '17 at 14:42
@DaveL.Renfro I strongly agree with you. – José Carlos Santos Jul 10 '17 at 14:43
@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. – Chris Jul 10 '17 at 17:48
Other suggestions please? – Chris Jul 11 '17 at 13:42
3

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. – Dave L. Renfro Jul 11 '17 at 14:43
1

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. – Dave L. Renfro Jul 11 '17 at 14:51
My friend, thank you for all these useful information. Any other comment or suggestion (at any time) is welcome! – Chris Jul 12 '17 at 23:19

score 3 · Answer 2 · edited Feb 03 '22 at 01:13

3

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://coopersnotes.net/third_galois.html

edited Feb 03 '22 at 01:13

Bysshed

1,943

answered Jul 18 '17 at 07:26

nilo de roock

2,855

1

Thank you for your post. These notes are very useful (and with answers in exercises)! – Chris Jul 18 '17 at 08:38
Link is broken and I can't find it anywhere on the internet. – cqfd Jan 22 '20 at 11:08
@ThomasShelby I'll see if I can find my copy next Monday. – nilo de roock Jan 25 '20 at 19:33

score 2 · Answer 3 · answered Jul 18 '17 at 07:37

2

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.

answered Jul 18 '17 at 07:37

k.stm

19,187

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! – Chris Jul 18 '17 at 08:36
There is no English version. But yes, that book is awesome. – Martin Brandenburg Dec 24 '24 at 01:02
1

@MartinBrandenburg As a side note, I think that artificial intelligence has become so advanced by now that I suspect it can already be trusted to translate undergraduate textbooks on mathematics with some editorial help by a professional mathematician who is then left with doing only some 1-10% of the work for the translation. Perhaps that’s a worthwhile endeavour … – k.stm Jan 06 '25 at 10:56
1

@MartinBrandenburg And as another example, I have your book on category theory here. You said you specifically chose German for it because there is a shortage of German undergraduate literature on the subject, but I think it would also serve as a unique and very worthy addition to the English literature. Moreover, it would be nice to finally have full translation of EGA to English. I really think that already o1 could serve this purpose. And if it isn’t o1, I still have hopes for o3 or some of the future models. – k.stm Jan 06 '25 at 10:59
@MartinBrandenburg Just to let you know (and everyone who may read this comment), there is an English translation of Bosch's Algebra since 2018. – user926356 Apr 08 '25 at 02:56
@user926356 Funny. That means that I was wrong when I wrote my answer and then Martin was wrong when he wrote his comment. – k.stm Apr 09 '25 at 10:35

Galois Theory books (in association with Abstract Algebra books)

3 Answers3

Linked