---
date: 2022-01-23
---

# 2022-01-23

Tags:
#untagged #web/quick-notes 

Refs:
?

## 18:36

- What is the [zeta function](zeta%20function) of a [number field](number%20field.md)?
- Idea: $\ZZ$ and $\FF_q[t]$ are similar.
  Quotients by maximal ideals are fields, while the characteristics are mixed for $\ZZ$ and all $p$ for the latter.
- Dirichlet's theorem: for each $m\in \ZZ_{\geq 1}$ and $a$ coprime to $m$, there are infinite many primes $p\equiv a\mod m$.
  - By Chebotarev density, for a fixed modulus $m$ the primes are equidistributed among residue classes.

- The absolute value on $\QQ$ inherited from $\RR$ is the infinite place $\abs{\wait}_\infty$.
The finite places are the \(p\dash \)adic valuations $v_p$.

- Recovering a module as an intersection of its local points:

![](figures/2022-01-23_19-39-56.png)

![](figures/2022-01-23_19-40-45.png)

  - Important strategy: to show a property holds for an ideal $I \normal R$ an integral domain, show that it holds for all of its localizations $I\plocalize{\mfp}$ for $\mfp \in \spec R$, and that the property is preserved under intersections.