Editing Algebraic number theory (section)

===Local fields===
{{Main|Local field}}
[[Completion (metric space)|Completing]] a number field ''K'' at a place ''w'' gives a [[complete field]]. If the valuation is Archimedean, one obtains '''R''' or '''C''', if it is non-Archimedean and lies over a prime ''p'' of the rationals, one obtains a finite extension <math>K_w/\mathbf{Q}_p:</math> a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the [[Kronecker–Weber theorem]] can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.