# 2022-01-23

Refs: ?

## 18:36

• What is the zeta function of a number field?

• Idea: $${\mathbf{Z}}$$ and $${ \mathbf{F} }_q[t]$$ are similar. Quotients by maximal ideals are fields, while the characteristics are mixed for $${\mathbf{Z}}$$ and all $$p$$ for the latter.

• Dirichlet’s theorem: for each $$m\in {\mathbf{Z}}_{\geq 1}$$ and $$a$$ coprime to $$m$$, there are infinite many primes $$p\equiv a\operatorname{mod}m$$.

• By Chebotarev density, for a fixed modulus $$m$$ the primes are equidistributed among residue classes.
• The absolute value on $${\mathbf{Q}}$$ inherited from $${\mathbf{R}}$$ is the infinite place $${\left\lvert {{-}} \right\rvert}_\infty$$. The finite places are the $$p{\hbox{-}}$$adic valuations $$v_p$$.

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

• Important strategy: to show a property holds for an ideal $$I {~\trianglelefteq~}R$$ an integral domain, show that it holds for all of its localizations $$I \left[ { \scriptstyle { { ({{\mathfrak{p}}}^c) }^{-1}} } \right]$$ for $${\mathfrak{p}}\in \operatorname{Spec}R$$, and that the property is preserved under intersections.
#untagged #web/quick-notes