Tags: #untagged #web/quick-notes
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.