Today on the message boards at work, I saw someone asking a question about describing \(\text{Spec}(\mathbb{Z}[x])\) as a topological space. The person asking the question either had no idea what he was talking about or else I have forgotten way too much about algebra because he was alluding to the orbits of an action of a Galois group, and I was thinking that this could not possibly be the right way to go about doing this because Galois groups are nice when you have groups and/or fields, and this does not sound at all like a problem about groups and/or fields. It doesn’t even seem like a problem about modules, so there really isn’t anything groupy about it.

While I may have forgotten a lot about ring theory, I am pretty sure that when I am 90 years old and living in some sort of institution and no longer remember the names of my relatives, I will still remember that the most important reason that \(\mathbb{Z}[x]\) shows up in problems is because it is not a principal ideal domain, which means that the prime ideals are not necessarily maximal ideals.

If this question had been about the Zariski topology of the prime spectrum of polynomials over an algebraically closed field, then it would have been trivial. Prime ideals are maximal, irreducible polynomials are linear, yawn.

But since I had actual work to do and since we fastidiously avoid giving away the answer when we help students out on the message boards (also I think that this student was possibly cheating on homework), I did not bother to remind myself of what happens when you construct this topology. It’s pretty easy to classify the prime ideals of \(\mathbb{Z}[x]\). And I decided that I no longer care what exactly the closed sets are. Would one want to define an affine variety? Do something with Hilbert’s Nullstellensatz? Not me.

So I steered the student away from his (likely) ill-fated voyage through Galois groups and his potential detours through something scary and cohomological. Someone else posted something that was definitely related but was scheme-y enough that I decided to ignore it.

Good luck, student who is possibly cheating on homework! If you are not cheating on homework, then you are going to have a lot better luck getting a good answer on StackExchange, where the people answering questions remember a lot more algebra than I do!