This is a place to collect resources and advice that I or others I know have found helpful for learning algebra.
Overview (skip this section if you just want specific links and references)
Part of what makes algebra a difficult subject is that there is a large breadth and depth of content, with each area requiring its own notions and skills. Roughly, the main content areas of algebra are as follows.
- Group theory
- Ring theory
- Module theory
- Field theory
- Galois theory
- Representation theory of finite groups
Algebra courses are conventionally taught in a two-semester format. The first semester usually focuses on group theory, ring theory, and a bit of field theory. The second semester usually focuses on module theory, field theory, and Galois theory. The representation theory of finite groups is a common supplemental topic and a good jumping off point for learning about the wonderful world of representation theory.
Most math PhD programs require the above as foundational knowledge for all of their students. Of course, there’s more to algebra than what appears above. Three closely related areas are:
- Category theory
- Homological algebra
- Commutative algebra
Category theory should be viewed as a broad organizational framework for all of math (except for maybe (real) analysis and low-dimensional topology). It is based on the idea that in math we often want to study objects (commonly sets with extra structure) and certain maps between them (commonly functions that preserve the extra structure). An excellent example is the category of groups, whose “objects” are groups and maps or “morphisms” are group homomorphisms. We also often want to compare different categories, an idea captured by the notion of a functor. A functor is like a generalized function that tells you what to do with objects and morphisms between those objects. One example is the “forgetful” functor, which goes from the category of groups to the category of sets and “forgets” that groups and group homomorphisms have extra structure. In other words, it sees that a group is a set with some (blah) and forgets (blah).
One of the big advantages of category theory is that it helps us “get under the hood” of algebra. Something you will see repeatedly as you learn algebra is that the same ideas keep popping up. Examples include kernels, quotients, and “freeness” (think basis of a vector space). Category theory tells us what things are true about these ideas in general (i.e., independent of context), and thus helps us identify when something more than “abstract nonsense” is going on. I often think about this in terms of going on a trip. Category theory guarantees you can get from point A to point B once you put some fuel in the tank. How you put that fuel in the tank, and whether you have any fuel in the first place, is something specific to each category. More formally, you can always write down some abstract formalism to describe a desired property or trait. If said thing exists then it will always behave the same way. But it may not always exist…
Homological algebra and commutative algebra are more direct offshoots of algebra. Homological algebra helps answer questions of the following flavor. Given a linear map between vector spaces that is not surjective, “how far” is it from being surjective? Ditto for replacing ‘surjective’ by ‘injective.’ It turns out that such questions are ubiquitous in math. Commutative algebra, on the other hand, is specifically interested in the study of commutative rings. All of the ideas from ring theory — e.g., ideals and principal ideal domains (PIDs) — make appearances here, alongside new ideas like localization and maximal ideals. Commutative algebra is super important because it helps us model phenomena in number theory, algebraic geometry, and other areas.
Advice
Algebra has a wide variety of languages and perspectives. One facet of this is an abundance of terms all used to describe the same thing. Examples include:
- Stabilizer = isotropy group (in group theory)
- Uniformizer = local coordinate (in algebraic number theory)
- Span of = vector space generated by (in linear algebra)
Taking in all of this at once, as well as swallowing the generally high level of abstraction, can be overwhelming. A good way to deal with this is to build intuitive models of things. Create a narrative in your head (or a spoken narrative, if you can find a willing victim) with all the key ideas and players that avoids the technicalities as much as possible. Or map things out in a big diagram. Things that have a story to them are things you can understand and recall accurately. This is how most mathematicians remember a lot of math.
Another good idea is to learn things several times over. In the short term this involves reading from a variety of sources and integrating the elements of each that work best for you. Some sources deliver good intuition. Other sources provide all the details. Still other sources give lots of examples. What you need in order to work with something will depend on how you learn and what you need to do. Your approach to a concept will vary considerably between trying to digest a chunk of theory and trying to explicitly do a computation. Authors will approach their mathematical exposition with their own ambitions. Don’t expect your motivations and aims to always match with those of any given author.
In the longer term you might consider looking at sources that provide a “second course” (or even third+) on a given topic. Note that many authors will bill their work as a first course even though it may be exceedingly more ambitious than other first course texts. There are many reasons for this, though undoubtedly a large and common one is that seasoned experts often underestimate the struggle of learning things for the first time. There’s often a lot of controversy and debate (especially among educators) about how to learn and present things for the first time. And for good reason. Your first shot at learning algebra is likely to be a little messy and have some holes. It’s okay to be pragmatic about things and move forward, so long as you return when you feel the timing is right.
Resources
At this point you’ve probably listened to me ramble for long enough and want some concrete suggestions. So, let’s do it, starting with some algebra book recommendations.
- Abstract Algebra by David Dummit and Richard Foote
This book was my first course on abstract algebra. It’s a big book that’s pretty comprehensive for foundation. The authors are very chatty, which I find charming but those who prefer more terse exposition might find annoying or unnecessary. One downside of the amount of text is that things can sometimes be a little hard to find on the page. This problem is not the same across all printings, so your mileage may vary. Another advantage of this book is that it has a lot of exercises. They’re not necessarily all good — longer exercises can be too hand-holdy and difficulty sometimes spikes without clear indication — but there is enough variety to suit a first-pass through the material.
- Algebra: Chapter 0 by Paolo Aluffi
This book is billed as a first course but should be treated as a second. The aim is to understand abstract algebra in a categorical context. This book does a good job of landing somewhere reasonable between Dummit and Foote and resources like nlab (look it up if you’re unfamiliar). One advantage is once again the plethora of exercises. Quality of exercises is generally pretty high, and the exercise sections flesh out a good stock of examples in addition to the core text. Bonus points for a solid treatment of homological algebra that does not have a million typos and errata.
- Linear Algebra Done Right by Sheldon Axler
I haven’t actually read this one but have heard good things about it. The idea is to do linear algebra without determinants. This will definitely help you with the theoretical aspects of being an amateur algebraist. One issue with the text is that determinants are extremely important in other areas of math, even at an abstract level and even in areas very close to algebra. Also, you need determinants if you want to do any computations! With this in mind, a good complementary text I have heard recommended is Linear Algebra Done Wrong by Sergei Treil.
These papers were my go-to resource when I was learning algebra for the first time. Keith has a wonderful style of exposition — especially the flow and aesthetic sense — that resonate with me. Lots of detail and so many worked examples. Much more than just algebra too.
Tai-Danae Bradley’s excellent picture-based math blog. Includes a solid treatment of things outside of algebra too.
Qiaochu Yuan’s blog. There’s lots of neat stuff on here (much of which I haven’t read). Particular recommendations relevant to algebra include his treatment of the Sylow theorems and group actions in general.