Galois Theory?

Hey everyone, I hope all is well. I am looking for book recommendations for Galois Theory, I have completed both Ring/Field Theory and Group Theory, but my university won't allow me to take Galois Theory, which is their Graduate Algebra class, so I decided to do a self study.

[davidshechtman]

I've always wanted a shirt that says: "The Galios Group, Providing semetric solutions to polynomial problems since 1832"

[卡喇邦噶男]

I think Jacobson’s Basic Algebra I is nice and comprehensive, Emil Artin’s little book on Galois Theory is also commonly praised. If you hope to learn infinite Galois theory, you can turn to Pierre Guillot - “A Gentle Course in Local Class Field Theory_ Local Number Fields, Brauer Groups, Galois Cohomology”. I personally think finite Galois theory is easier and very elegant, the infinite one requires some back ground in topological group.

[antikertech]

I learned a small section on the most important results of Galois Theory in university. It's not difficult to understand once you are familiar about rings, fields, field extensions, etc.
Personally, I don't like to learn from textbooks unless I have no other choice, instead I rather learn from the translated original paper. Learning from the masters of the past gives you a unique opportunity for growth in your mathematical insight. I got myself an english translation of the entire work of Bernhard Riemann, I am learning complex analysis from the master himself.
Since you like languages, there are three languages that will serve you the most in your mathematical study: german, french, and academic latin. I already began learning german and I've been practicing it almost everyday.

[spenxerbdp]

Lang algebra introduces Galois theory nicely

[midwesteigensheaf]

I took a course on Galois theory this past spring, and we used Dummit and Foote. I wholeheartedly recommend D&F as a source of exercises, but it may not be the best for self study. I really enjoyed Jean-Pierre Escofier’s “Galois Theory, ” which starts with the history of the general solutions of the quadratic, cubic, and quartic equations, then gets into a deeper study of polynomials, field extensions, and so on. I loved the exercises, and the book has solutions to many of the exercises at the end of each chapter which is helpful for self study. I also liked Richard Borcherds’ Galois theory lecture series on youtube to supplement reading

[suranami]

I used Edwards and occasionally looked at Milne's lecture notes. Galois theory is hard to wrap your head around but fairly applicable in many real-world situations. Evan Chen's napkin also has good notes on it. Serge Lang's "Algebra" also has a chapter on it with pretty great exercises imo. They are hard, though (in fact, the whole book is).

[akrishna]

My favorite book was by Stewart, supplemented by Cox and some fantastic notes on Dr. Keith Conrad's page of expository texts. I didn't find it extremely difficult, but as with rings & fields it takes some time to get used to certain terminology & recognize their importance (e.g. when are extensions algebraic, simple, separable, normal). I will say that I found myself "getting" Galois theory much better in the context of algebraic number theory. It's a lot of fun, honestly - seeing how all sorts of algebraic objects tie together elegantly. (Edit: I definitely second the Stacks project chapter on fields, and D&F is always useful for exercises)

[thegroosh]

There’s also Galois theory by Cox. It’s not really suited for a quick study and it doesn’t go that deep, for instance there is no Galois theory of infinite extensions. However, it contains an incredible amount of historical context and many explicitly computed examples. It’s a beautiful book

[anonymoose]

Algebraic number theory by Neukirch and the Stacks Project's chapter on fields.

[chergle]

I took an algebra course that covered Galois theory my last semester of undergrad. It was a course that totally clicked for me -- field and Galois theory are really beautiful. I would second the recommendation you saw for Dummit & Foote.

As for how hard it is, it's going to be harder or easier largely depending on how solid your group theory is. There's some technical definitions on the field theory side too (hard to keep track of things like normal/separable field extensions), but being able to prove/disprove the existence of certain (normal) subgroups of finite groups that you know is crucial. I would suggest reviewing that if you're about to dive into Galois theory.

[matthiaspfau]

Milne notes for free on his site. It's condensed yet has good examples and expands on computational aspects.
Neukirch and like texts focus on rings of integers in numberfields, which uses Galois theory but focus on other topics.

[lucascardoso]

Hi there. The lecture notes by Tom Leinster are very nice. I find it simple and fun, perfect for self-study. What do you intend to study next?

[virat.chauhan]

Algebra (From the viewpoint of Galois Theory) by Siegfried Bosch

[pauselab]

also trying to learn galois theory. took me almost a year now to learn about all the groups, rings, fields extension etc...

[abhinashsharma]

Galois theory by Ian Stewart... Kindly have a look 😊🌈👍

[margaritawithoutastraw]

Hi! I used Morandi, it's pretty good. It's hard alright, but I'm an algebra kind of person, so putting in effort and rereading the text usually does it.