Working Through Jacobson When You Actually Need It
Nathan Jacobson's Basic Algebra II Second Edition is one of those books that sits on a shelf looking intimidating until you actually open it. I picked it up around 2018 because my graduate qualifying exam had a section on ring theory and field extensions that nobody had properly prepared me for. The first edition came out in the seventies, the second edition filled in some gaps but didn't change the fundamental character of the thing. It is dense. It expects you to already know what you are doing. Academic publishers rarely put this kind of material behind paywalls that make sense. Dover Publications carries the second edition and sells it for around twenty dollars in paperback. That is the version most people actually use. The original Interscience/Wiley hardcover runs somewhere between eighty and two hundred depending on whether you are buying it used from a seller who thinks they are selling a collectible. Amazon, AbeBooks, and BookScouter will have listings. If you need it for a course, check if your library has a copy on reserve before spending money on it. The ISBN for the Dover second edition is 978-0486471885. Sometimes you will see that number swapped around by sellers who list it incorrectly. Double check before you buy. The book is structured around four main parts: polynomial rings and factorization, linear algebra over fields, Galois theory, and then some supplementary chapters on modules and commutative rings. Part one alone takes up roughly a third of the volume and covers what most people would call advanced undergraduate algebra. If you already know basic abstract algebra from Dummit and Foote or Fraleigh, Jacobson will feel repetitive in places and frustratingly terse in others. That is normal. The terseness is deliberate. He assumes you can fill in the gaps yourself.
How to actually read it without losing a week
Start with the exercises at the end of each section. Not after you finish the chapter, before you move on. Jacobson writes proofs that skip steps deliberately, and the exercise set is where he hides the things he does not want to prove inline. I spent two full days stuck on Exercise 14 in Section 4.2 of Chapter 5 because the text never actually explains why a certain ideal is prime in the polynomial ring over a finite field. The answer is in the hint, but the hint assumes you have already proven a lemma three sections back. Once I went back and proved that lemma myself, the exercise took five minutes. Here is the practical approach I ended up using consistently. Read the theorem statement carefully. Skip the proof on the first pass. Go to the exercise set and attempt the first three problems. If you cannot solve them, now go read the proof. You will understand what the author was actually trying to accomplish instead of just following symbolic manipulation. This reverses the normal reading order and feels wrong at first. It works because Jacobson structures his material so that the examples only make sense once you understand why the theorem matters. The linear algebra sections are where the book becomes genuinely useful. Most algebra texts treat vector spaces as an afterthought. Jacobson treats them as the foundation. Chapter 3 runs through the theory of modules over principal ideal domains with enough detail that you can use it directly for classification of finitely generated abelian groups. I used this exact section to prep for a comprehensive exam question on rational canonical form. Nobody in my program had covered it properly in the coursework. Jacobson gave me everything I needed in about sixty pages.
What the book gets wrong or leaves out
There are gaps. The second edition added some material but did not fix everything. The treatment of computational algebra is essentially nonexistent. If you need to actually compute a Gröbner basis or perform row reduction over a large finite field, you are on your own. There are occasional typographical errors that propagate through the exercises. In the 1989 Dover printing, Exercise 7 in Chapter 6, Section 3 has a sign error in the statement of the polynomial. It does not affect the result but it will waste you twenty minutes if you do not catch it. I caught it by working through the example problem that precedes it and noticing the discrepancy. The Galois theory chapters assume familiarity with field extensions that some readers will not have. Jacobson mentions separability and normality in passing without building up the intuition from first principles. If you are encountering these concepts for the first time, supplement with chapters four and five of Ian Stewart's Galois Theory before returning to Jacobson. The contrast between Stewart's pedagogical approach and Jacobson's compression will make the material click faster than either book alone.
Get the Full Details
Real problem I ran into and how I worked around it
I was preparing for a qualifying exam and needed to verify that a specific quotient ring was a field. The polynomial in question was degree seven over Z/3Z. Jacobson's approach would be to check irreducibility using the factorization tests in Section 2.4, but those tests require computing gcds with x3k x for various k. Doing this by hand is brutal. I spent about forty-five minutes on the first gcd and made an arithmetic error. Once I found a mistake, I had to start over because one wrong coefficient cascades through the entire calculation. The workaround was to use a computer algebra system for the mechanical part and then verify the logic by hand. I ran the irreducibility check in SageMath in about twelve seconds, then went back to Jacobson's proof technique to confirm that the output was correct. The actual learning value was in reconstructing the argument, not in performing the computation. This is the pattern I repeated throughout the book: let machines do the arithmetic, use Jacobson for the structure.
Who should actually use this book
It is not a self-study book in the traditional sense. If you are working through it alone, plan on spending significantly more time per page than a standard textbook. The exercise set ranges from trivial verification to problems that require genuine insight. The harder problems sometimes connect to material that Jacobson never explicitly teaches. I found myself referencing Mac Lane and Birkhoff's Algebra and Artin's Algebra alongside it constantly. None of those references replace Jacobson, but they make the reading process tolerable. If you are taking a graduate algebra course, this book is excellent as a secondary reference. If you are trying to learn the subject for the first time from scratch, start elsewhere and return to Jacobson when you need deeper coverage of a specific topic. The second edition is still in print and worth the Dover price. The older Wiley editions are not worth the premium unless you have a specific reason to own them.
