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## The arithmetic plane

If you haven’t seen them already, you might want to read John Baez’s week205 and Lieven le Bruyn’s series of posts on the subject of spectra. I especially recommend that you take a look at the picture of $\text{Spec } \mathbb{Z}[x]$ to which Lieven le Bruyn links before reading this post. John Baez’s introduction to week205 would probably also have served as a great introduction to this series before I started it:

There’s a widespread impression that number theory is about numbers, but I’d like to correct this, or at least supplement it. A large part of number theory – and by the far the coolest part, in my opinion – is about a strange sort of geometry. I don’t understand it very well, but that won’t prevent me from taking a crack at trying to explain it….

Before we talk about localization again, we need some examples of rings to localize. Recall that our proof of the description of $\text{Spec } \mathbb{C}[x, y]$ also gives us a description of $\text{Spec } \mathbb{Z}[x]$:

Theorem: $\text{Spec } \mathbb{Z}[x]$ consists of the ideals $(0), (f(x))$ where $f$ is irreducible, and the maximal ideals $(p, f(x))$ where $p \in \mathbb{Z}$ is prime and $f(x)$ is irreducible in $\mathbb{F}_p[x]$.

The upshot is that we can think of the set of primes of a ring of integers $\mathbb{Z}[\alpha] \simeq \mathbb{Z}[x]/(f(x))$, where $f(x)$ is a monic irreducible polynomial with integer coefficients, as an “algebraic curve” living in the “plane” $\text{Spec } \mathbb{Z}[x]$, which is exactly what we’ll be doing today. (When $f$ isn’t monic, unfortunate things happen which we’ll discuss later.) We’ll then cover the case of actual algebraic curves next.