Question

Prerequisites to understanding Galois Theory.

Answer 1

Please tell us your mathematical background if you want to get a useful answer. There's a big difference in the books that would be relevant to you if you have already seen basic finite group theory (homomorphisms, quotient groups, etc) and ring theory (ring homomorphisms, ideals in K[x] for a field K, etc) compared to having seen no abstract algebra at all.

Answer 2

I think the main prerequisite for the theory of field extensions is linear algebra. Next, I’d say finite group theory. Some ring theory is good to know (like that k[x] is a PID for a field k) but for the most part it’s just linear algebra and group theory.

Emil Artin has a Galois theory book which starts with a quick refresher of linear algebra. Dummit and Foote is a very comprehensive book and I certainly don’t think you need everything from the earlier chapters to understand the field theory ones. In fact, I’d be very hesitant to say I’ve mastered all the content before then and i haven’t had an issue learning basic Galois theory.

Answer 3

Assuming Linear Algebra background: 1,2,3,7,8,9,13,14

Group Theory and Linear Algebra background: 7,8,9,13,14

Answer 4

I recently watched a video on YouTube recommending Edwards' Galois theory. It seems to start from the basics and building only the necessary to understand GT. I haven't read it yet so I can't give a personalized recommendation.

Answer 5

Most parts of Galois theory is self-contained, so personally you only need to know about vector spaces, bases and a little about homomorphism. I also recommend reading lecture notes instead of books since these notes is usually written in a more comprehensive ways compare to books and it is free. For eg: notes of Miles Reid on GT

Answer 6

Undergraduate algebra should be enough to start looking at an introduction text at the upper undergrad/early grad level.

Answer 7

The 4th edition of Ian Stewart's *Galois Theory* offers a terse but thoughtful introduction, I highly recommend it. Also, plenty of history is included if you're into that. As far as prerequisites go, a decent understanding of basic Ring theory will be helpful. Some of the first important technical results you encounter, namely Artin's extension Theorem and isomorphisms of splitting fields, are quite dependent on polynomial rings, their quotients, and ring homomorphisms. Knowledge of linear algebra, often of finite-dimensional vector spaces, is needed for building up field extensions. Lastly, I would suggest being comfortable with the idea of a group (the symmetric group in particular), normal subgroups, and the First Isomorphism Theorem for groups. I've found that some of the more niche group theory concepts, like solvability, are often introduced in tandem with Galois Theory making it quite digestible.

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Galois Theory is a remarkable chunk of math, I'm excited for you. Enjoy!

Answer 8

If you're looking at reading Dummit and Foote I would reccomend reading Chapters 1 and 2, Sections 3.1-3.3, Chapter 4, Section 5.1, Section 6.1 (you specifically should read about Solvable groups), Sections 7.1-7.4, Chapter 8, Sections 9.1-9.4, and pretty much all of Chapter 14. If you don't feel confident with Linear Algebra, you might want to review that as well.

I don't have much experience with other texts, but I wouldn't reccomend using one that doesn't treat finite group theory, the basics of commutative ring theory (specifically polynomial rings), and the theory of field extensions before diving into Galois Theory. You really need to understand these topics in order to get a good understanding of Galois Theory.

Answer 9

I highly recommend the beautiful book 'Galois Theory' by Harold M. Edwards. It takes you on a very thorough (at the undergraduate level) but gentle course through Galois Theory (including its motivations and history) in only \~100 pages! It is fully self-contained (if you know what a field and a vector space is, you meet the prerequisites).

Answer 10

If you already know some stuff about rings, ideals, groups and fields (meaning you took some undergraduate course in abstract Álgebra), take a look at Pierre Samuel Algebraic Theory of Numbers. It's a cheap Dover book, only about 100 pages long. It introduces algebraic number theory, is beautifully written, and finishes with a chapter on Galois Theory