2022-01-23

Tags: #untagged #web/quick-notes

Refs: ?

18:36

  • What is the zeta function of a number field?

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

  • Dirichlet’s theorem: for each \(m\in {\mathbb{Z}}_{\geq 1}\) 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 \({\mathbb{Q}}\) inherited from \({\mathbb{R}}\) 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