In Valuations and Completions we introduced the $p$-adic numbers $\mathbb{Q}_{p}$, which, like the real numbers, are the completion of the rational numbers under a certain kind of valuation. There is one such valuation for each prime number $p$, and another for the “infinite prime”, which is just the usual absolute value. Each valuation may be thought of as encoding number theoretic information related to the prime $p$, or to the “infinite prime”, for the case of the absolute value (more technically, the $p$-adic valuations are referred to as nonarchimedean valuations, while the absolute value is an example of an archimedean valuation).

We can consider valuations not only for the rational numbers, but for more general algebraic number fields as well. In its abstract form, given an algebraic number field $K$, a (multiplicative) valuation of $K$ is simply any function $|\ |$ from $K$ to $\mathbb{R}$ satisfying the following properties:

(i) $|x|\geq 0$, where $x=0$ if and only if $x=0$

(ii) $|xy|=|x||y|$

(iii) $|x+y|\leq|x|+|y|$

If this seems reminiscent of the discussion in Metric, Norm, and Inner Product, it is because a valuation does, in fact, define a metric on $K$, and by extension, a topology. Two valuations are equivalent if they define the same topology; another way to phrase this statement is that two valuations $|\ |_{1}$ and $|\ |_{2}$ are equivalent if $|x|_{1}=|x|_{2}^{s}$ for some positive real number $s$, for all $x\in K$.  The valuation is nonarchimedean if $|x+y|\leq\text{max}\{|x|,|y|\}$; otherwise, it is archimedean.

Just as in the case of rational numbers, we also have an exponential valuation, defined as a function $v$ from the field $K$ to $\mathbb{R}\cup \infty$ satisfying the following conditions:

(i) $v(x)=\infty$ if and only if $x=0$

(ii) $v(xy)=v(x)+v(y)$

(iii) $v(x+y)\geq\text{min}\{v(x),v(y)\}$

Two exponential valuations $v_{1}$ and $v_{2}$ are equivalent if $v_{1}(x)=sv_{2}(x)$ for some real number $s$, for all $x\in K$.

The idea of valuations allows us to make certain concepts in algebraic number theory (see Algebraic Numbers) more abstract. We define a place $v$ of an algebraic number field $K$ as an equivalence class of valuations of $K$. We write $K_{v}$ to denote the completion of $K$ under the place $v$; these are the generalizations of the $p$-adic numbers and real numbers to algebraic number fields other than $\mathbb{Q}$. The nonarchimedean places are also called the finite places, while the archimedean places are also called the infinite places. To express whether a place $v$ is a finite place or an infinite place, we write $v|\infty$ or $v\nmid\infty$ respectively.

The infinite places are of two kinds; the ones for which $K_{v}$ is isomorphic to $\mathbb{R}$ are called the real places, while the ones for which $K_{v}$ is isomorphic to $\mathbb{C}$ are called the complex places. The number of real places and complex places of $K$, denoted by $r_{1}$ and $r_{2}$ respectively, satisfy the equation $r_{1}+2r_{2}=n$, where $n$ is the degree of $K$ over $\mathbb{Q}$, i.e. $n=[K:\mathbb{Q}]$.

By the way, in some of the literature, such as in the book Algebraic Number Theory by Jurgen Neukirch, “places” are also referred to as “primes“. This is intentional – one may actually think of our definition of places as being like a more abstract replacement of the definition of primes. This is quite advantageous in driving home the concept of primes as equivalence classes of valuations; however, to avoid confusion, we will stick to using the term “places” here, along with its corresponding notation.

When $v$ is a nonarchimedean valuation, we let $\mathfrak{o}_{v}$ denote the set of all elements $x$ of $K_{v}$ for which $|x|_{v}\leq 1$. It is an example of a ring with special properties called a valuation ring. This means that, for any $x$ in $K$, either $x$ or $x^{-1}$ must be in $\mathfrak{o}_{v}$. We let $\mathfrak{o}_{v}^{*}$ denote the set of all elements of $\mathfrak{o}_{v}$ for which $|x|_{v}=1$, and we let $\mathfrak{p}_{v}$ denote the set of all elements of $\mathfrak{o}_{v}$ for which $|x|_{v}< 1$. It is the unique maximal ideal of $\mathfrak{o}_{v}$.

Now we proceed to consider the modern point of view in algebraic number theory, which is to consider all these equivalence classes of valuations together. This will lead us to the language of adeles and ideles.

An adele $\alpha$ of $K$ is a family $(\alpha_{v})$ of elements $\alpha_{v}$ of $K_{v}$ where $\alpha_{v}\in K_{v}$, and $\alpha_{v}\in\mathfrak{o}_{v}$ for all but finitely many $v$. We can define addition and multiplication componentwise on adeles, and the resulting ring of adeles is then denoted $\mathbb{A}_{K}$. The group of units of the ring of adeles is called the group of ideles, denoted $I_{K}$. For a finite set of primes $S$ that includes the infinite primes, we let

$\displaystyle \mathbb{A}_{K}^{S}=\prod_{v\in S}K_{v}\times\prod_{v\notin S}\mathfrak{o}_{v}$

and

$\displaystyle I_{K}^{S}=\prod_{v\in S}K_{v}^{*}\times\prod_{v\notin S}\mathfrak{o}_{v}^{*}$.

We denote the set of infinite primes by $S_{\infty}$. Then $\mathfrak{o}_{K}$, the ring of integers of the number field $K$, is given by $K\cap\mathbb{A}_{K}^{S_{\infty}}$, while $\mathfrak{o}_{K}^{*}$, the group of units of $\mathfrak{o}_{K}$, is given by $K^{*}\cap I_{K}^{S_{\infty}}$.

Any element of $K$ is also an element of $\mathbb{A}_{K}$, and any element of $K^{*}$ (the group of units of $K$) is also an element of $I_{K}$. The elements of $I_{K}$ which are also elements of $K^{*}$ are called the principal ideles. This should not be confused with the concept of principal ideals; however the terminology is perhaps suggestive on purpose. In fact, ideles and fractional ideals are related. Any fractional ideal $\mathfrak{a}$ can be expressed in the form

$\displaystyle \mathfrak{a}=\prod_{\mathfrak{p}}\mathfrak{p}^{\nu_{\mathfrak{p}}}$.

Therefore, we have a mapping

$\displaystyle \alpha\mapsto (\alpha)=\prod_{\mathfrak{p}}\mathfrak{p}^{v_{\mathfrak{p}}(\alpha_v)}$

from the group of ideles to the group of fractional ideals. This mapping is surjective, and its kernel is $I_{K}^{S_{\infty}}$.

The quotient group $I_{K}/K^{*}$ is called the idele class group of $K$, and is denoted by $C_{K}$. Again, this is not to be confused with the ideal class group we discussed in Algebraic Numbers, although the two are related; in the language of ideles, the ideal class group is defined as $I_{K}/I_{K}^{S_{\infty}}K^{*}$, and is denoted by $Cl_{K}$. There is a surjective homomorphism $C_{K}\mapsto Cl_{K}$ induced by the surjective homomorphism from the group of ideles to the group of fractional ideals that we have described in the preceding paragraph.

An important aspect of the concept of adeles and ideles is that they can be equipped with topologies (see Basics of Topology and Continuous Functions). For the adeles, this topology is generated by the neighborhoods of $0$ in $\mathbb{A}_{K}^{S_{\infty}}$ under the product topology. For the ideles, this topology is defined by the condition that the mapping $\alpha\mapsto (\alpha,\alpha^{-1})$ from $I_{K}$ into $\mathbb{A}_{K}\times\mathbb{A}_{K}$ be a homeomorphism onto its image. Both topologies are locally compact, which means that every element has a neighborhood which is compact, i.e. every open cover of that neighborhood has a finite subcover. For the group of ideles, its topology is compatible with its group structure, which makes it into a locally compact topological group.

In this post, we have therefore seen how the theory of valuations can allow us to consider a more abstract viewpoint for algebraic number theory, and how considering all the valuations together to form adeles and ideles allows us to rephrase the usual concepts related to algebraic number fields, such as the ring of integers, its group of units, and the ideal class group, in a new form. In addition, the topologies on the adeles and ideles can be used to obtain new results; for instance, because the group of ideles is a locally compact topological (abelian) group, we can use the methods of harmonic analysis (see Some Basics of Fourier Analysis) to study it. This is the content of the famous thesis of the mathematician John Tate. Another direction where the concept of adeles and ideles can take us is class field theory, which relates the idele class group to the other important group in algebraic number theory, the Galois group (see Galois Groups). The language of adeles and ideles can also be applied not only to algebraic number fields but also to function fields of curves over finite fields. Together these fields are also known as global fields.

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