Algebraic number field

Prerequisites

The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication. A prominent example of a field is the field of rational numbers, commonly denoted ${\displaystyle \mathbb {Q} }$, together with its usual operations of addition and multiplication.

Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)

${\displaystyle (x_{1},x_{2},\dots )}$

whose entries are elements of a fixed field, such as the field ${\displaystyle \mathbb {Q} }$. Any two such sequences can be added by adding the corresponding entries. Furthermore, all members of any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces may be of infinite length. If, however, the vector space consists of finite sequences

${\displaystyle (x_{1},\dots ,x_{n})}$,

the vector space is said to be of finite dimension, ${\displaystyle n}$.

Definition

An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over ${\displaystyle \mathbb {Q} }$.

Non-examples

• The real numbers, ${\displaystyle \mathbb {R} }$, and the complex numbers, ${\displaystyle \mathbb {C} }$, are fields that have infinite dimension as ${\displaystyle \mathbb {Q} }$-vector spaces; hence, they are not number fields. This follows from the uncountability of ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {C} }$ as sets, whereas every number field is necessarily countable.
• The set ${\displaystyle \mathbb {Q} ^{2}}$ of ordered pairs of rational numbers, with the entry-wise addition and multiplication is a two-dimensional commutative algebra over ${\displaystyle \mathbb {Q} }$. However, it is not a field, since it has zero divisors: $${\displaystyle (1,0)\cdot (0,1)=(1\cdot 0,0\cdot 1)=(0,0)}$$

Unique factorization

For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal ${\displaystyle (6)}$ in the ring ${\displaystyle \mathbf {Z} [{\sqrt {-5}}]}$ of quadratic integers factors into prime ideals as

${\displaystyle (6)=(2,1+{\sqrt {-5}})(2,1-{\sqrt {-5}})(3,1+{\sqrt {-5}})(3,1-{\sqrt {-5}})}$

However, unlike ${\displaystyle \mathbf {Z} }$ as the ring of integers of ${\displaystyle \mathbf {Q} }$, the ring of integers of a proper extension of ${\displaystyle \mathbf {Q} }$ need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in ${\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}=\mathbf {Z} [{\sqrt {-5}}]}$, the uniqueness of the factorization fails:

${\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})\cdot (1-{\sqrt {-5}})}$

Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in ${\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}}$. Euclidean domains are unique factorization domains; for example ${\displaystyle \mathbf {Z} [i]}$, the ring of Gaussian integers, and ${\displaystyle \mathbf {Z} [\omega ]}$, the ring of Eisenstein integers, where ${\displaystyle \omega }$ is a cube root of unity (unequal to 1), have this property.^[1]

Analytic objects: ζ-functions, L-functions, and class number formula

The failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of the so-called ideal class group. This group is always finite. The ring of integers ${\displaystyle {\mathcal {O}}_{K}}$ possesses unique factorization if and only if it is a principal ring or, equivalently, if ${\displaystyle K}$ has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields (i.e., ${\displaystyle \mathbf {Q} ({\sqrt {-d}}),d\geq 1}$) with prescribed class number. The class number formula relates h to other fundamental invariants of ${\displaystyle K}$. It involves the Dedekind zeta function ${\displaystyle \zeta _{K}(s)}$, a function in a complex variable ${\displaystyle s}$, defined by

${\displaystyle \zeta _{K}(s):=\prod _{\mathfrak {p}}{\frac {1}{1-N({\mathfrak {p}})^{-s}}}.}$

(The product is over all prime ideals of ${\displaystyle {\mathcal {O}}_{K}}$, ${\displaystyle N({\mathfrak {p}})}$ denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field ${\displaystyle {\mathcal {O}}_{K}/{\mathfrak {p}}}$. The infinite product converges only for Re(s) > 1; in general analytic continuation and the functional equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ_{${\displaystyle \mathbb {Q} }$}(s) = ζ(s).

The class number formula states that ζ_{${\displaystyle K}$}(s) has a simple pole at s = 1 and at this point the residue is given by

${\displaystyle {\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot h\cdot \operatorname {Reg} }{w\cdot {\sqrt {|D|}}}}.}$

Here r₁ and r₂ classically denote the number of real embeddings and pairs of complex embeddings of ${\displaystyle K}$, respectively. Moreover, Reg is the regulator of ${\displaystyle K}$, w the number of roots of unity in ${\displaystyle K}$ and D is the discriminant of ${\displaystyle K}$.

Dirichlet L-functions ${\displaystyle L(\chi ,s)}$ are a more refined variant of ${\displaystyle \zeta (s)}$. Both types of functions encode the arithmetic behavior of ${\displaystyle \mathbb {Q} }$ and ${\displaystyle K}$, respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression

${\displaystyle a,a+m,a+2m,\ldots }$

with coprime ${\displaystyle a}$ and ${\displaystyle m}$, there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet ${\displaystyle L}$-function is nonzero at ${\displaystyle s=1}$. Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.^[2]

Integral basis

An integral basis for a number field ${\displaystyle K}$ of degree ${\displaystyle n}$ is a set

B = {b₁, …, b_n}

of n algebraic integers in ${\displaystyle K}$ such that every element of the ring of integers ${\displaystyle {\mathcal {O}}_{K}}$ of ${\displaystyle K}$ can be written uniquely as a Z-linear combination of elements of B; that is, for any x in ${\displaystyle {\mathcal {O}}_{K}}$ we have

x = m₁b₁ + ⋯ + m_nb_n,

where the m_i are (ordinary) integers. It is then also the case that any element of ${\displaystyle K}$ can be written uniquely as

m₁b₁ + ⋯ + m_nb_n,

where now the m_i are rational numbers. The algebraic integers of ${\displaystyle K}$ are then precisely those elements of ${\displaystyle K}$ where the m_i are all integers.

Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.

Power basis

Let ${\displaystyle K}$ be a number field of degree ${\displaystyle n}$. Among all possible bases of ${\displaystyle K}$ (seen as a ${\displaystyle \mathbb {Q} }$-vector space), there are particular ones known as power bases, that are bases of the form

${\displaystyle B_{x}=\{1,x,x^{2},\ldots ,x^{n-1}\}}$

for some element ${\displaystyle x\in K}$. By the primitive element theorem, there exists such an ${\displaystyle x}$, called a primitive element. If ${\displaystyle x}$ can be chosen in ${\displaystyle {\mathcal {O}}_{K}}$ and such that ${\displaystyle B_{x}}$ is a basis of ${\displaystyle {\mathcal {O}}_{K}}$ as a free Z-module, then ${\displaystyle B_{x}}$ is called a power integral basis, and the field ${\displaystyle K}$ is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial^[3] $${\displaystyle x^{3}-x^{2}-2x-8.}$$

Example

Consider the field extension ${\displaystyle \mathbb {Q} (\theta )}$ where ${\displaystyle \theta =\zeta _{3}{\sqrt[{3}]{2}}}$. Then, we have a ${\displaystyle \mathbb {Q} }$-basis given by $${\displaystyle \{1,\zeta _{3}{\sqrt[{3}]{2}},\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}\}}$$ since any ${\displaystyle x\in \mathbb {Q} (\theta )}$ can be expressed as some ${\displaystyle \mathbb {Q} }$-linear combination $${\displaystyle a+b\zeta _{3}{\sqrt[{3}]{2}}+c\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}=a+b\theta +c\theta ^{2}}$$ Then, we can take some ${\displaystyle y\in \mathbb {Q} (\theta )}$ where ${\displaystyle y=y_{0}+y_{1}\theta +y_{2}\theta ^{2}}$ and compute ${\displaystyle x\cdot y}$. Writing this out gives $${\displaystyle {\begin{aligned}a(y_{0}+y_{1}\theta +y_{2}\theta ^{2})+\\b(2y_{2}+y_{0}\theta +y_{1}\theta ^{2})+\\c(2y_{1}+2y_{2}\theta +y_{0}\theta ^{2})\end{aligned}}}$$ We can find the matrix ${\displaystyle A(x)}$ by writing out the associated matrix equation giving $${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}{\begin{bmatrix}y_{0}\\y_{1}\\y_{2}\end{bmatrix}}={\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\by_{0}+ay_{1}+2cy_{2}\\cy_{0}+by_{1}+ay_{2}\end{bmatrix}}}$$ showing $${\displaystyle A(x)={\begin{bmatrix}a&2c&2b\\b&a&2c\\c&b&a\end{bmatrix}}}$$ We can then compute the trace and determinant with relative ease, giving the trace and norm.

Properties

By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linear function of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(λx) = λ Tr(x), and the norm is a multiplicative homogeneous function of degree n: N(xy) = N(x) N(y), N(λx) = λⁿ N(x). Here λ is a rational number, and x, y are any two elements of ${\displaystyle K}$.

The trace form derived is a bilinear form defined by means of the trace, as $${\displaystyle Tr_{K/L}:K\otimes _{L}K\to L}$$ by $${\displaystyle Tr_{K/L}(x\otimes y)=Tr_{K/L}(x\cdot y)}$$${\displaystyle {\text{Tr}}_{K/L}(x)}$. The integral trace form, an integer-valued symmetric matrix is defined as ${\displaystyle t_{ij}={\text{Tr}}_{K/\mathbb {Q} }(b_{i}b_{j})}$, where b₁, ..., b_n is an integral basis for ${\displaystyle K}$. The discriminant of ${\displaystyle K}$ is defined as det(t). It is an integer, and is an invariant property of the field ${\displaystyle K}$, not depending on the choice of integral basis.

The matrix associated to an element x of ${\displaystyle K}$ can also be used to give other, equivalent descriptions of algebraic integers. An element x of ${\displaystyle K}$ is an algebraic integer if and only if the characteristic polynomial p_A of the matrix A associated to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the Cayley–Hamilton theorem, p_A(A) = 0, and it follows that p_A(x) = 0, so that x is an algebraic integer. Conversely, if x is an element of ${\displaystyle K}$ that is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrix in a suitable basis of ${\displaystyle K}$. The property of being an algebraic integer is defined in a way that is independent of a choice of a basis in ${\displaystyle K}$.

Example with integral basis

Consider ${\displaystyle K=\mathbb {Q} (x)}$, where x satisfies x³ − 11x² + x + 1 = 0. Then an integral basis is [1, x, 1/2(x² + 1)], and the corresponding integral trace form is $${\displaystyle {\begin{bmatrix}3&11&61\\11&119&653\\61&653&3589\end{bmatrix}}.}$$

The "3" in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of ${\displaystyle K}$ on ${\displaystyle K}$. This basis element induces the identity map on the 3-dimensional vector space, ${\displaystyle K}$. The trace of the matrix of the identity map on a 3-dimensional vector space is 3.

The determinant of this is 1304 = 2³·163, the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is 5216 = 2⁵·163.

Examples

The field ${\displaystyle K=\mathbb {Q} [x]/(x^{6}-2)=\mathbb {Q} (\theta )}$ for ${\displaystyle \theta =\zeta {\sqrt[{6}]{2}}}$ where ${\displaystyle \zeta }$ 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^{[6]pg 15-16}.

Archimedean places

Here we use the standard notation ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ for the number of real and complex embeddings used, respectively (see below).

Calculating the archimedean places of a number field ${\displaystyle K}$ is done as follows: let ${\displaystyle x}$ be a primitive element of ${\displaystyle K}$, with minimal polynomial ${\displaystyle f}$ (over ${\displaystyle \mathbb {Q} }$). Over ${\displaystyle \mathbb {R} }$, ${\displaystyle f}$ will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots ${\displaystyle r}$ of factors of degree one are necessarily real, and replacing ${\displaystyle x}$ by ${\displaystyle r}$ gives an embedding of ${\displaystyle K}$ into ${\displaystyle \mathbb {R} }$; the number of such embeddings is equal to the number of real roots of ${\displaystyle f}$. Restricting the standard absolute value on ${\displaystyle \mathbb {R} }$ to ${\displaystyle K}$ gives an archimedean absolute value on ${\displaystyle K}$; such an absolute value is also referred to as a real place of ${\displaystyle K}$. On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into ${\displaystyle \mathbb {C} }$. Either one of this pair of embeddings can be used to define an absolute value on ${\displaystyle K}$, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of ${\displaystyle K}$.^[7][8]

If all roots of ${\displaystyle f}$ above are real (respectively, complex) or, equivalently, any possible embedding ${\displaystyle K\subseteq \mathbb {C} }$ is actually forced to be inside ${\displaystyle \mathbb {R} }$ (resp. ${\displaystyle \mathbb {C} }$), ${\displaystyle K}$ is called totally real (resp. totally complex).^[9][10]

Non-Archimedean or ultrametric places

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

For any ultrametric place v we have that |x|_v ≤ 1 for any x in ${\displaystyle {\mathcal {O}}_{K}}$, since the minimal polynomial for x has integer factors, and hence its p-adic factorization has factors in Z_p. 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 O_K

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

|x + y|_v ≤ max (|x|_v, |y|_v) < 1.

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

Conversely, given a prime ideal ${\displaystyle {\mathfrak {p}}}$ of ${\displaystyle {\mathcal {O}}_{K}}$, a discrete valuation can be defined by setting ${\displaystyle v_{\mathfrak {p}}(x)=n}$ where n is the biggest integer such that ${\displaystyle x\in {\mathfrak {p}}^{n}}$, 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 ${\displaystyle K}$ correspond to prime ideals of ${\displaystyle {\mathcal {O}}_{K}}$. For ${\displaystyle K=\mathbb {Q} }$, 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 ${\displaystyle {\mathcal {O}}_{K}}$. Given an ultrametric place ${\displaystyle v}$ on a number field ${\displaystyle K}$, the corresponding localization is the subring ${\displaystyle T}$ of ${\displaystyle K}$ of all elements ${\displaystyle x}$ such that | x |_v ≤ 1. By the ultrametric property ${\displaystyle T}$ is a ring. Moreover, it contains ${\displaystyle {\mathcal {O}}_{K}}$. For every element x of ${\displaystyle K}$, at least one of x or x⁻¹ is contained in ${\displaystyle T}$. Actually, since K^×/T^× can be shown to be isomorphic to the integers, ${\displaystyle T}$ is a discrete valuation ring, in particular a local ring. Actually, ${\displaystyle T}$ is just the localization of ${\displaystyle {\mathcal {O}}_{K}}$ at the prime ideal ${\displaystyle {\mathfrak {p}}}$, so ${\displaystyle T={\mathcal {O}}_{K,{\mathfrak {p}}}}$. Conversely, ${\displaystyle {\mathfrak {p}}}$ is the maximal ideal of ${\displaystyle T}$.

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

An example

The following example illustrates the notions introduced above. In order to compute the ramification index of ${\displaystyle \mathbb {Q} (x)}$, where

f(x) = x³ − x − 1 = 0,

at 23, it suffices to consider the field extension ${\displaystyle \mathbb {Q} _{23}(x)/\mathbb {Q} _{23}}$. Up to 529 = 23² (i.e., modulo 529) f can be factored as

f(x) = (x + 181)(x² − 181x − 38) = gh.

Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation | y |_g for y given by g is | −191 |₂₃ = 1. On the other hand, the same substitution in h yields y² − 161y − 161 modulo 529. Since 161 = 7 × 23,

${\displaystyle \left\vert y\right\vert _{h}={\sqrt {\left\vert 161\right\vert }}_{23}={\frac {1}{\sqrt {23}}}}$

Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.

The valuations of any element of ${\displaystyle K}$ can be computed in this way using resultants. If, for example y = x² − x − 1, using the resultant to eliminate x between this relationship and f = x³ − x − 1 = 0 gives y³ − 5y² + 4y − 1 = 0. If instead we eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both 1 in this instance.)

Dedekind discriminant theorem

Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in ${\displaystyle \mathbb {Q} _{p}}$ where p divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime p divides the discriminant, then there is a p-place that ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field ${\displaystyle \mathbb {Q} (x)}$ with x³ − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does. The other ramified place comes from the absolute value on the complex embedding of ${\displaystyle K}$.

Local and global fields

Number fields share a great deal of similarity with another class of fields much used in algebraic geometry known as function fields of algebraic curves over finite fields. An example is K_p(T). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient fields of which are the function fields in question) of curves. Therefore, both types of field are called global fields. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding local fields. For number fields ${\displaystyle K}$, the local fields are the completions of ${\displaystyle K}$ at all places, including the archimedean ones (see local analysis). For function fields, the local fields are completions of the local rings at all points of the curve for function fields.

Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case.

Hasse principle

A prototypical question, posed at a global level, is whether some polynomial equation has a solution in ${\displaystyle K}$. If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of ${\displaystyle K}$, which is often easier, since analytic methods (classical analytic tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not hold, however, for more general types of equations. However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completions K_v can be explicitly determined, whereas the Galois groups of global fields, even of ${\displaystyle \mathbb {Q} }$ are far less understood.

Adeles and ideles

In order to assemble local data pertaining to all local fields attached to ${\displaystyle K}$, the adele ring is set up. A multiplicative variant is referred to as ideles.

Generalizations

• Algebraic function field

Algebraic number theory

• Dirichlet's unit theorem, S-unit
• Kummer extension
• Minkowski's theorem, Geometry of numbers
• Chebotarev's density theorem

Class field theory

• Ray class group
• Decomposition group
• Genus field