Editing P-adic number (section)

== Generalizations and related concepts ==
The reals and the {{mvar|p}}-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general [[algebraic number field]]s, in an analogous way. This will be described now.

Suppose ''D'' is a [[Dedekind domain]] and ''E'' is its [[field of fractions]]. Pick a non-zero [[prime ideal]] ''P'' of ''D''. If ''x'' is a non-zero element of ''E'', then ''xD'' is a [[fractional ideal]] and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of ''D''. We write ord<sub>''P''</sub>(''x'') for the exponent of ''P'' in this factorization, and for any choice of number ''c'' greater than 1 we can set
:<math>|x|_P = c^{-\!\operatorname{ord}_P(x)}.</math>
Completing with respect to this absolute value {{nowrap begin}}|⋅|<sub>''P''</sub>{{nowrap end}} yields a field ''E''<sub>''P''</sub>, the proper generalization of the field of ''p''-adic numbers to this setting. The choice of ''c'' does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the [[residue field]] ''D''/''P'' is finite, to take for ''c'' the size of ''D''/''P''.

For example, when ''E'' is a [[number field]], [[Ostrowski's theorem]] says that every non-trivial [[absolute value (algebra)|non-Archimedean absolute value]] on ''E'' arises as some {{nowrap begin}}|⋅|<sub>''P''</sub>{{nowrap end}}. The remaining non-trivial absolute values on ''E'' arise from the different embeddings of ''E'' into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of ''E'' into the fields '''C'''<sub>''p''</sub>, thus putting the description of all
the non-trivial absolute values of a number field on a common footing.)

Often, one needs to simultaneously keep track of all the above-mentioned completions when ''E'' is a number field (or more generally a [[global field]]), which are seen as encoding "local" information. This is accomplished by [[adele ring]]s and [[idele group]]s.

''p''-adic integers can be extended to [[Solenoid (mathematics)#p-adic solenoids|''p''-adic solenoids]] <math>\mathbb{T}_p</math>. There is a map from <math>\mathbb{T}_p</math> to the [[circle group]] whose fibers are the ''p''-adic integers <math>\mathbb{Z}_p</math>, in analogy to how there is a map from <math>\mathbb{R}</math> to the circle whose fibers are <math>\mathbb{Z}</math>.