Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number. This situation changed with the discovery of
p-adic numbers by
Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field K into its various topological
completions K_{\mathfrak{p}} at once. A
place of a number field K is an equivalence class of
absolute values on Kpg 9. Essentially, an absolute value is a notion to measure the size of elements x of Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). The equivalence relation between absolute values |\cdot|_0 \sim |\cdot|_1 is given by some \lambda \in \mathbb{R}_{>0} such that|\cdot|_0 = |\cdot|_1^{\lambda}meaning we take the value of the norm |\cdot|_1 to the \lambda-th power. In general, the types of places fall into three regimes. Firstly (and mostly irrelevant), the trivial absolute value | |0, which takes the value 1 on all non-zero The second and third classes are
Archimedean places and
non-Archimedean (or ultrametric) places. The completion of K with respect to a place |\cdot|_{\mathfrak{p}} is given in both cases by taking
Cauchy sequences in Kand dividing out
null sequences, that is, sequences \{x_n\}_{n \in \mathbb{N}} such that |x_n|_\mathfrak{p} \to 0tends to zero when n tends to infinity. This can be shown to be a field again, the so-called completion of K at the given place {{nowrap||\cdot|_\mathfrak{p},}} denoted {{nowrap|K_{\mathfrak{p}}.}} For {{nowrap|K = \mathbb{Q},}} the following non-trivial norms occur (
Ostrowski's theorem): the (usual)
absolute value, sometimes denoted |\cdot|_\infty, which gives rise to the complete
topological field of the real numbers {{nowrap|\mathbb{R}.}} On the other hand, for any prime number p, the
p-adic absolute value is defined by :|
q|
p =
p−
n, where
q =
pn a/
b and
a and
b are integers not divisible by
p. It is used to construct the p-adic numbers {{nowrap|\mathbb{Q}_p.}} In contrast to the usual absolute value, the
p-adic absolute value gets
smaller when
q is multiplied by
p, leading to quite different behavior of \mathbb{Q}_p as compared to {{nowrap|\mathbb{R}.}} Note the general situation typically considered is taking a number field K and considering a
prime ideal \mathfrak{p} \in \text{Spec}(\mathcal{O}_K) for its associated
ring of algebraic numbers {{nowrap|\mathcal{O}_K.}} Then, there will be a unique place |\cdot|_{\mathfrak{p}}: K \to \mathbb{R}_{\geq 0} called a non-Archimedean place. In addition, for every embedding \sigma: K \to \mathbb{C} there will be a place called an Archimedean place, denoted {{nowrap||\cdot|_\sigma:K\to \mathbb{R}_{\geq 0}.}} This statement is a theorem also called
Ostrowski's theorem.
Examples The field K = \mathbb{Q}[x]/(x^6 - 2) = \mathbb{Q}(\theta) for \theta = \zeta\sqrt[6]{2} where \zeta is a fixed 6th root of unity, provides a rich example for constructing explicit real and complex Archimedean embeddings, and non-Archimedean embeddings as well If all roots of f above are real (respectively, complex) or, equivalently, any possible embedding K \subseteq \mathbb{C} is actually forced to be inside \mathbb{R} (resp. not be inside {{nowrap|\mathbb{R}),}} K is called
totally real (resp.
totally complex).
Non-Archimedean or ultrametric places To find the non-Archimedean places, let again f and x be as above. In {{nowrap|\mathbb{Q}_p ,}} f splits in factors of various degrees, none of which are repeated, and the degrees of which add up to the degree of For each of these p-adically irreducible factors we may suppose that x satisfies f_i and obtain an embedding of K into an algebraic extension of finite degree over {{nowrap|\mathbb{Q}_p.}} Such a
local field behaves in many ways like a number field, and the 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 {{nowrap|\mathbb{Q}_p .}} By using this p-adic norm map N_{f_i} for the place f_i, we may define an absolute value corresponding to a given p-adically irreducible factor f_i of degree m by|y|_{f_i} = |N_{f_i}(y)|_p^{1/m}Such an absolute value is called an
ultrametric, non-Archimedean or p-adic place of For any ultrametric place
v we have that |
x|
v ≤ 1 for any
x in {{nowrap|\mathcal{O}_K,}} since the minimal polynomial for
x has integer factors, and hence its
p-adic factorization has factors in
Zp. 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.
Prime ideals in OK For an ultrametric place
v, the subset of \mathcal{O}_K defined by |
x|
v \mathfrak{p} of {{nowrap|\mathcal{O}_K.}} This relies on the ultrametricity of
v: given
x and
y in {{nowrap|\mathfrak{p},}} then :|
x +
y|
v ≤ max (|
x|
v, |y|
v) \mathfrak{p} is even a
prime ideal. Conversely, given a prime ideal \mathfrak{p} of {{nowrap|\mathcal{O}_K,}} a
discrete valuation can be defined by setting v_\mathfrak{p}(x) = n where
n is the biggest integer such that {{nowrap|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 correspond to prime ideals of {{nowrap| \mathcal{O}_K.}} For {{nowrap|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 {{nowrap|\mathcal{O}_K.}} Given an ultrametric place v on a number field the corresponding localization is the subring T of K of all elements x such that |
x |
v ≤ 1. By the ultrametric property T is a ring. Moreover, it contains {{nowrap|\mathcal{O}_K.}} For every element
x of at least one of
x or
x−1 is contained in Actually, since
K×/
T× can be shown to be isomorphic to the integers, T is a
discrete valuation ring, in particular a
local ring. Actually, T is just the localization of \mathcal{O}_K at the prime ideal {{nowrap|\mathfrak{p},}} so {{nowrap|T = \mathcal{O}_{K,\mathfrak{p}}.}} Conversely, \mathfrak{p} is the maximal ideal of Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.
Lying over theorem and places Some of the basic theorems in algebraic number theory are the
going up and going down theorems, which describe the behavior of some prime ideal \mathfrak{p} \in \text{Spec}(\mathcal{O}_K) when it is extended as an ideal in \mathcal{O}_L for some field extension We say that an ideal \mathfrak{o} \subset \mathcal{O}_L
lies over \mathfrak{p} if {{nowrap|\mathfrak{o}\cap\mathcal{O}_K = \mathfrak{p}.}} Then, one incarnation of the theorem states a prime ideal in \text{Spec}(\mathcal{O}_L) lies over {{nowrap|\mathfrak{p},}} hence there is always a surjective map\text{Spec}(\mathcal{O}_L) \to \text{Spec}(\mathcal{O}_K)induced from the inclusion {{nowrap|\mathcal{O}_K \hookrightarrow \mathcal{O}_L.}} Since there exists a correspondence between places and prime ideals, this means we can find places dividing a place that is induced from a field extension. That is, if p is a place of then there are places v of L that divide in the sense that their induced prime ideals divide the induced prime ideal of p in {{nowrap|\text{Spec}(\mathcal{O}_L).}} In fact, this observation is usefulpg 129-131; hence\begin{align} K\otimes_\mathbb{Q}\mathbb{Q}_p &\cong \frac{ \mathbb{Q}_p[X] }{ \prod_{v|p}Q_v(X) } \\&\cong \bigoplus_{v|p}K_v \end{align}Moreover, there are embeddings\begin{align} i_v:&K \to K_v \\ & \theta \mapsto \theta_v \end{align}where \theta_v is a root of Q_v giving K_v = \mathbb{Q}_p(\theta_v); hence we could writeK_v = i_v(K)\mathbb{Q}_p as subsets of \mathbb{C}_p (which is the completion of the algebraic closure of {{nowrap| \mathbb{Q}_p).}} ==Ramification==