structural:scholar_edition, linguistic:paraphrasing, information:classification, linguistic:translation
Non-Archimedean or ultrametric places

To find the non-Archimedean places, let again f {\displaystyle f} and x {\displaystyle x} be as above. In Q p {\displaystyle \mathbb {Q} _{p}}, f {\displaystyle f} splits in factors of various degrees, none of which are repeated, and the degrees of which add up to n {\displaystyle n}, the degree of f {\displaystyle f}. For each of these p {\displaystyle p} -adically irreducible factors f i {\displaystyle f_{i}}, we may suppose that x {\displaystyle x} satisfies f i {\displaystyle f_{i}} and obtain an embedding of K {\displaystyle K} into an algebraic extension of finite degree over Q p {\displaystyle \mathbb {Q} _{p}}. Such a local field behaves in many ways like a number field, and the p {\displaystyle p} -adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to Q p {\displaystyle \mathbb {Q} _{p}}. By using this p {\displaystyle p} -adic norm map N f i {\displaystyle N_{f_{i}}} for the place f i {\displaystyle f_{i}}, we may define an absolute value corresponding to a given p {\displaystyle p} -adically irreducible factor f i {\displaystyle f_{i}} of degree m {\displaystyle m} by | y | f i = | N f i (y) | p 1 / m {\displaystyle |y|_{f_{i}}=|N_{f_{i}}(y)|_{p}^{1/m}} Such an absolute value is called an ultrametric, non-Archimedean or p {\displaystyle p} -adic place of K {\displaystyle K}.

For any ultrametric place v we have that | x | ≤ 1 for any x in O K {\displaystyle {\mathcal {O}}_{K}}, since the minimal polynomial for x has integer factors, and hence its p -adic factorization has factors in Z. Consequently, the norm term (constant term) for each factor is a p -adic integer, and one of these is the integer used for defining the absolute value for v.