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## Z[sqrt{-3}] is the Eisenstein integers glued together at two points

Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by $\text{Spec } R$ I mean a commutative ring $R$ regarded as an object in the opposite category $\text{CRing}^{op}$.

In the previous post we showed that the splitting behavior of a rational prime $p$ in the ring of cyclotomic integers $\mathbb{Z}[\zeta_n]$ depends only on the residue class of $p \bmod n$. This is suggestive enough of quadratic reciprocity that now would be a good time to give a full proof.

The key result is the following fundamental observation.

Proposition: Let $q$ be an odd prime. Then $\mathbb{Z}[\zeta_q]$ contains $\sqrt{ q^{*} } = \sqrt{ (-1)^{ \frac{q-1}{2} } q}$.

Quadratic reciprocity has a function field version over finite fields which David Speyer explains the geometric meaning of in an old post. While this is very much in line with what we’ve been talking about, it’s a little over my head, so I’ll leave it for the interested reader to peruse.

## 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.

## Primes and ideals

Probably the first important result in algebraic number theory is the following. Let $K$ be a finite field extension of $\mathbb{Q}$. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$.

Theorem: The ideals of $\mathcal{O}_K$ factor uniquely into prime ideals.

This is the “correct” generalization of the fact that $\mathbb{Z}$, as well as some small extensions of it such as $\mathbb{Z}[ \imath ], \mathbb{Z}[\omega]$, have unique factorization of elements. My goal, in this next series of posts, is to gain some intuition about this result from a geometric perspective, wherein one thinks of a ring as the ring of functions on some space. Setting up this perspective will take some time, and I want to do it slowly. Let’s start with the following simple questions.

• What is the right notion of prime?
• Why look at ideals instead of elements?

In this series I will assume the reader is familiar with basic abstract algebra but may not have a strong intuition for it.