Any recommendations for good resources for learning Galois Theory?

[dx]

Summary:: recommend learning materials

Although I took galois theory as an undergraduate, I never really studied it. Any recommendations for good resources?

 

[fresh_42]

I have read Artin and van der Waerden, but whether these are suited for you depends on your taste, more than it depends on the sources. How do you learn best? Videos? Books? Bourbaki stylish books? Lectures? Lecture notes?

 

[dx]

I have read Artin and van der Waerden, but whether these are suited for you depends on your taste, more than it depends on the sources. How do you learn best? Videos? Books? Bourbaki stylish books? Lectures? Lecture notes?

I prefer books, notes and so on. I tend to not be able to pay much attention during live lectures, so I prefer books. I like the Bourbaki style.

 

[Demystifier]

Bourbaki stylish books?

What are the names of other styles? Russian style? Is there any other?

 

[fresh_42]

What are the names of other styles? Russian style? Is there any other?

Good question. I tend to say pre- and post-Bourbaki era. Or the comparison lexicon versus essay could work. I had to think about Kurosh as you said Russian style. I have his group theory books, and yes, it is definitely pre-Bourbaki. However, I think it is more a matter of times than it is a matter of locations. Artin's Galoistheory is similar, and I don't think Artin wanted to write Russian style, neither intended van der Waerden.

It would be interesting to figure out what drove this development. The need to learn more in less time? The possibility to quickly lookup things? Or simply the wish to order mathematics by the authors of Bourbaki after Hilbert's second problem failed to be solvable.

 

[fresh_42]

I prefer books, notes and so on. I tend to not be able to pay much attention during live lectures, so I prefer books. I like the Bourbaki style.

... in which case Serge Lang is the answer to your question:
https://www.amazon.com/dp/038795385X/?tag=pfamazon01-20

But there are also
https://www.amazon.com/dp/B000QEIU16/?tag=pfamazon01-20
https://www.amazon.com/dp/0387987657/?tag=pfamazon01-20
which deal directly with Galois theory.

These are books for the shelves. If you are only interested in the content, then it is probably cheaper to search the internet for lecture notes, e.g. via Google and the search key 'Galois theory pdf'. My results were:

https://www.jmilne.org/math/CourseNotes/FT.pdf (recommended!)
https://pages.uoregon.edu/koch/Galois.pdf

 

[mathwonk]

It may depend in part on how much background you retain. As a student I also never really learned Galois theory, and as a professor it always seemed to get squeezed out the end of the courses I taught and we never got there. So one year I resolved to teach Galois theory first, so I could learn it myself. That was not so easy as I realized there is so much background material, groups, linear algebra, ring and field theory. So I spent the first quarter mostly on group theory, then snuck in a little linear algebra (dimension theory and determinants), then finished up the first quarter by discussing finite field extensions, the concept of normal extensions, and finally at least defining a Galois group, and using it to prove that a solvable polynomial must have a solvable Galois group. That took about 125 pages of class notes. (843-1 and 843-2 on my webpage.)

the second quarter we looked at the converse result, that in characteristic zero, a polynomial with a solvable Galois group in fact has a solution formula by radicals, and we derived these formulas to some extent for cubics and quartics, with some errors in the case of quartics. Then we did some examples of calculations of Galois groups, and considered the inverse Galois problem: which finite groups occur as Galois groups, proving that all finite abelian groups do occur. The second quarter class notes occupied another 100+ pages. (844-1 and 844-2 on my webpage.)

This means that if you need to review the background algebra first, then do the Galois theory, and then also want to explore some of the applications, as I do in my notes, then at least my treatment of these topics took over 225 pages. My notes are free on my webpage, but unfortunately the quality of reproduction of the type fonts online has deteriorated over the decades since they were first posted, and they are a little troublesome to read now, but still readable with patience.

https://www.math.uga.edu/directory/people/roy-smith

I also did not understand the subject as well as the other authors referred to in previous posts here, but did my best to explain every detail to my class as well as I could. I.e. I read and tried to learn the material from the other more authoritative sources I cite, and then I tried to explain it in more detail than those sources did, which was my excuse for writing another book. I used and especially liked the book by Michael Artin, as opposed to the famous Notre Dame notes of his father Emil Artin, and would recommend Mike's book to almost anyone. It was written for sophomores at MIT. In fact if I had found Mike's book sooner, I might not have written my book. Our books do differ though since I include multilinear algebra in mine. I also happen to like Van der Waerden. Another book that is considered very user friendly is that of Ian Stewart. From the reviews on Amazon, I would suggest getting an older edition, e.g. from abebooks:
https://www.abebooks.com/servlet/Se...lts&an=ian+stewart&tn=galois+theory&kn=&isbn=

If you like the Bourbaki style, then chapter 5 of Bourbaki's Algebra, is devoted to a full discussion of the theory of commutative fields, including Galois theory, in about 120 pages. Here is a reasonably priced french copy:
https://www.abebooks.com/servlet/BookDetailsPL?bi=30222162639&searchurl=bi=0&ds=30&bx=off&sortby=17&tn=algebre&an=bourbaki&recentlyadded=all&cm_sp=snippet-_-srp1-_-title17the only english translation i found was offered at over $2,000, but with free shipping!

 

[Infrared]

Here are some free online notes you may like: https://www.jmilne.org/math/CourseNotes/ft.html

 

[Demystifier]

For physicists (like me) who find the actual Galois theory too abstract, I found this:
http://www.cc.kyoto-su.ac.jp/project/MISC/slide/seminar-s/2011/120112Takeuchi.pdf

 

[dx]

Thanks for the recommendations. Will check them out. Looking around in my room, the only book I found which covers this topic is 'A survey of modern algebra' by garret birkhoff and saunders maclane. these are famous names, so maybe its ok.

Seems more approachable than the postgraduate level book by serge lang mentioned above, but i guess galois theory can be treated at various levels.

 

[WWGD]

What are the odds of joining an exchange here between members of the _42 family; fresh and love ?

 

[dx]

I also found this recently:

Lectures by Richard Borcherds. He also has various other courses on his youtube channel.