information:extraction
Prime ideals in O

For an ultrametric place v, the subset of O K {\displaystyle {\mathcal {O}}_{K}} defined by | x | < 1 is an ideal p {\displaystyle {\mathfrak {p}}} of O K {\displaystyle {\mathcal {O}}_{K}}. This relies on the ultrametricity of v: given x and y in p {\displaystyle {\mathfrak {p}}}, then

Actually, p {\displaystyle {\mathfrak {p}}} is even a prime ideal.

Conversely, given a prime ideal p {\displaystyle {\mathfrak {p}}} of O K {\displaystyle {\mathcal {O}}_{K}}, a discrete valuation can be defined by setting v p (x) = n {\displaystyle v_{\mathfrak {p}}(x)=n} where n is the biggest integer such that x ∈ p n {\displaystyle x\in {\mathfrak {p}}^{n}}, the n -fold power of the ideal. This valuation can be turned into an ultrametric place. Under this correspondence, (equivalence classes) of ultrametric places of K {\displaystyle K} correspond to prime ideals of O K {\displaystyle {\mathcal {O}}_{K}}. For K = Q {\displaystyle K=\mathbb {Q} }, this gives back Ostrowski's theorem: any prime ideal in Z (which is necessarily by a single prime number) corresponds to a non-Archimedean place and vice versa. However, for more general number fields, the situation becomes more involved, as will be explained below.

Yet another, equivalent way of describing ultrametric places is by means of localizations of O K {\displaystyle {\mathcal {O}}_{K}}. Given an ultrametric place v {\displaystyle v} on a number field K {\displaystyle K}, the corresponding localization is the subring T {\displaystyle T} of K {\displaystyle K} of all elements x {\displaystyle x} such that | x | ≤ 1. By the ultrametric property T {\displaystyle T} is a ring. Moreover, it contains O K {\displaystyle {\mathcal {O}}_{K}}. For every element x of K {\displaystyle K}, at least one of x or x is contained in T {\displaystyle T}. Actually, since K / T can be shown to be isomorphic to the integers, T {\displaystyle T} is a discrete valuation ring, in particular a local ring. Actually, T {\displaystyle T} is just the localization of O K {\displaystyle {\mathcal {O}}_{K}} at the prime ideal p {\displaystyle {\mathfrak {p}}}, so T = O K, p {\displaystyle T={\mathcal {O}}_{K,{\mathfrak {p}}}}. Conversely, p {\displaystyle {\mathfrak {p}}} is the maximal ideal of T {\displaystyle T}.

Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.