Number Theory: Local to Global...

In $\mathbb{Q}$, we often lack of tools to solve problems into itself. Its rings of integers $\mathbb{Z}$ have nice properties, for example: divisibility, factorization, ... But it is still complicated, for example, if $p\ne q\in Spec(\mathbb{Z})$, the residue fields $\mathbb{F}_q\ne \mathbb{F}_p$. Via localizing, $\mathbb{Z}$ looks nicer, the localization of $\mathbb{Z}$ at some prime $p\ne 0$ becomes a DVR.

For many number theory problems, we just want to look at one prime at a time, and that is why localization plays a role in this story. But sometimes algebraic properties is not enough, and one wants to use tools from analysis. But which kinds of analysis is suitable for arithmetic applications?  It is $p$-adic analysis. The first step is completing $\mathbb{Q}$, with $p$-adic norm, and then building theories for convergent, for series, and for finite extension of $\mathbb{Q}_p$. It is a new world, and very different from analysis in $\mathbb{R}$.

Things in local fields (of characteristic 0) are often much easier to understand (there are a lot of examples: there exists only finitely many extension of degree $n$ in a local field of characteristic 0, its Galois group (of finite Galois extension) is always solvable, and other interesting things in $p$-adic analysis). On the other hand, problems in global fields are often more difficult, and one does not have a "good enough" theory for global fields (some of the well-studied theories of number fields can be listed: Minkowski's theory (about lattices), cyclotomic fields, and ramification theory).

After having good things in local fields, one should think how to go back to number fields? The connection are various, it can be deduced via Galois groups and decomposition groups, via different theory, or via $p$-adic analysis.

One can also imitate and apply the theory of algebraic curves into number fields, i.e. by looking at "poles" and "zeros" of a "function" at all residue fields, and this leads to a version of Riemann-Roch's theorem for number fields, or even better, a Grothendieck-Riemann-Roch's theorem. This can be found in the book of Neukirch about Algebraic Number Theory.

Interesting stories now begin...