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Algebraic number field

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Finite degree (and hence algebraic) field extension of the field of rational numbers

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Inmathematics, analgebraic number field (or simplynumber field) is anextension field $K {\displaystyle K}$ of thefield ofrational numbers $\mathbb {Q}$ such that thefield extension $K/\mathbb {Q}$ hasfinite degree (and hence is analgebraic field extension).Thus $K {\displaystyle K}$ is a field that contains $\mathbb {Q}$ and has finitedimension when considered as avector space over $\mathbb {Q}$ .

The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic ofalgebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods.

Definition

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Prerequisites

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Main articles:Field andVector space

The notion of algebraic number field relies on the concept of afield. A field consists of aset of elements together with two operations, namelyaddition, andmultiplication, and somedistributivity 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 ofrational numbers, commonly denoted $\mathbb {Q}$ , together with its usual operations of addition and multiplication.

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

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

whose entries are elements of a fixed field, such as the field $\mathbb {Q}$ . Any two such sequences can be added by adding the corresponding entries. Furthermore, all members of any sequence can be multiplied by a single elementc of the fixed field. These two operations known asvector addition andscalar 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 offinite sequences

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

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

Definition

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Analgebraic number field (or simplynumber field) is a finite-degree field extension of the field of rational numbers. Heredegree means the dimension of the field as a vector space over $\mathbb {Q}$ .

Examples

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The smallest and most basic number field is the field $\mathbb {Q}$ of rational numbers. Many properties of general number fields are modeled after the properties of $\mathbb {Q}$ . At the same time, many other properties of algebraic number fields are substantially different from the properties of rational numbers—one notable example is that thering ofalgebraic integers of a number field is not aprincipal ideal domain, in general.
TheGaussian rationals, denoted $\mathbb {Q} (i)$ (read as " $\mathbb {Q}$ adjoined $i {\displaystyle i}$ "), form the first (historically) non-trivial example of a number field. Its elements are elements of the form $a+bi$ where botha andb are rational numbers andi is theimaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity $i^{2}=-1.$ Explicitly, for real numbers $a, b, c, d {\displaystyle a,b,c,d}$ : ${\begin{matrix}(a+bi)+(c+di)&=&(a+c)+(b+d)i\\(a+bi)\cdot (c+di)&=&(ac-bd)+(ad+bc)i\end{matrix}}$ Non-zero Gaussian rational numbers areinvertible, which can be seen from the identity $(a+bi)\left({\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i\right)={\frac {(a+bi)(a-bi)}{a^{2}+b^{2}}}=1.$ It follows that the Gaussian rationals form a number field that is two-dimensional as a vector space over $\mathbb {Q}$ .
More generally, for anysquare-free integer $d {\displaystyle d}$ , thequadratic field $\mathbb {Q} ({\sqrt {d}})$ is a number field obtained by adjoining the square root of $d {\displaystyle d}$ to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, $d=-1$ .
The cyclotomic field $\mathbb {Q} (\zeta _{n}),$ where $\zeta _{n}=\exp {(2\pi i/n)}$ , is a number field obtained from $\mathbb {Q}$ by adjoining a primitive $n {\displaystyle n}$ throot of unity $\zeta _{n}$ . This field contains all complex $n {\displaystyle n}$ th roots of unity and its dimension over $\mathbb {Q}$ is equal to $\varphi (n)$ , where $\varphi$ is theEuler totient function.

Non-examples

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Thereal numbers, $\mathbb {R}$ , and thecomplex numbers, $\mathbb {C}$ , are fields that have infinite dimension as $\mathbb {Q}$ -vector spaces; hence, they arenot number fields. This follows from theuncountability of $\mathbb {R}$ and $\mathbb {C}$ as sets, whereas every number field is necessarilycountable.
The set $\mathbb {Q} ^{2}$ ofordered pairs of rational numbers, with the entry-wise addition and multiplication is a two-dimensionalcommutative algebra over $\mathbb {Q}$ . However, it is not a field, since it haszero divisors: $(1,0)\cdot (0,1)=(1\cdot 0,0\cdot 1)=(0,0)$

Algebraicity, and ring of integers

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Generally, inabstract algebra, a field extension $K/L$ isalgebraic if every element $f {\displaystyle f}$ of the bigger field $K {\displaystyle K}$ is the zero of a (nonzero)polynomial with coefficients $e_{0},\ldots ,e_{m}$ in $L {\displaystyle L}$ :

p(f)=e_{m}f^{m}+e_{m-1}f^{m-1}+\cdots +e_{1}f+e_{0}=0

Every field extensionof finite degree is algebraic. (Proof: for $x {\displaystyle x}$ in $K {\displaystyle K}$ , simply consider $1,x,x^{2},x^{3},\ldots$ – we get a linear dependence, i.e. a polynomial that $x {\displaystyle x}$ is a root of.) In particular this applies to algebraic number fields, so any element $f {\displaystyle f}$ of an algebraic number field $K {\displaystyle K}$ can be written as a zero of a polynomial with rational coefficients. Therefore, elements of $K {\displaystyle K}$ are also referred to asalgebraic numbers. Given a polynomial $p {\displaystyle p}$ such that $p(f)=0$ , it can be arranged such that the leading coefficient $e_{m}$ is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as amonic polynomial. In general it will have rational coefficients.

If, however, the monic polynomial's coefficients are actually all integers, $f {\displaystyle f}$ is called analgebraic integer.

Any (usual) integer $z\in \mathbb {Z}$ is an algebraic integer, as it is the zero of the linear monic polynomial:

p(t)=t-z

It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of afinitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in $K {\displaystyle K}$ form aring denoted ${\mathcal {O}}_{K}$ called thering of integers of $K {\displaystyle K}$ . It is asubring of (that is, a ring contained in) $K {\displaystyle K}$ . A field contains nozero divisors and this property is inherited by any subring, so the ring of integers of $K {\displaystyle K}$ is anintegral domain. The field $K {\displaystyle K}$ is thefield of fractions of the integral domain ${\mathcal {O}}_{K}$ . This way one can get back and forth between the algebraic number field $K {\displaystyle K}$ and its ring of integers ${\mathcal {O}}_{K}$ . Rings of algebraic integers have three distinctive properties: firstly, ${\mathcal {O}}_{K}$ is an integral domain that isintegrally closed in its field of fractions $K {\displaystyle K}$ . Secondly, ${\mathcal {O}}_{K}$ is aNoetherian ring. Finally, every nonzeroprime ideal of ${\mathcal {O}}_{K}$ ismaximal or, equivalently, theKrull dimension of this ring is one. An abstract commutative ring with these three properties is called aDedekind ring (orDedekind domain), in honor ofRichard Dedekind, who undertook a deep study of rings of algebraic integers.

Unique factorization

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

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

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

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

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

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

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

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

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

The class number formula states that ζ_{$K {\displaystyle K}$}(s) has asimple pole ats = 1 and at this point theresidue is given by

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

Herer₁ andr₂ classically denote the number ofreal embeddings and pairs ofcomplex embeddings of $K {\displaystyle K}$ , respectively. Moreover, Reg is theregulator of $K {\displaystyle K}$ ,w the number ofroots of unity in $K {\displaystyle K}$ andD is the discriminant of $K {\displaystyle K}$ .

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

a,a+m,a+2m,\ldots

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

Bases for number fields

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Integral basis

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

B = {b₁, …,b_n}

ofn algebraic integers in $K {\displaystyle K}$ such that every element of the ring of integers ${\mathcal {O}}_{K}$ of $K {\displaystyle K}$ can be written uniquely as aZ-linear combination of elements ofB; that is, for anyx in ${\mathcal {O}}_{K}$ we have

x =m₁b₁ + ⋯ +m_nb_n,

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

m₁b₁ + ⋯ +m_nb_n,

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

Workinglocally and using tools such as theFrobenius map, it is always possible to explicitly compute such a basis, and it is now standard forcomputer algebra systems to have built-in programs to do this.

Power basis

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

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

for some element $x\in K$ . By theprimitive element theorem, there exists such an $x {\displaystyle x}$ , called aprimitive element. If $x {\displaystyle x}$ can be chosen in ${\mathcal {O}}_{K}$ and such that $B_{x}$ is a basis of ${\mathcal {O}}_{K}$ as a freeZ-module, then $B_{x}$ is called apower integral basis, and the field $K {\displaystyle K}$ is called amonogenic 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] $x^{3}-x^{2}-2x-8.$

Regular representation, trace and discriminant

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Recall that any field extension $K/\mathbb {Q}$ has a unique $\mathbb {Q}$ -vector space structure. Using the multiplication in $K {\displaystyle K}$ , an element $x {\displaystyle x}$ of the field $K {\displaystyle K}$ over the base field $\mathbb {Q}$ may be represented by $n\times n$ matrices $A=A(x)=(a_{ij})_{1\leq i,j\leq n}$ by requiring $xe_{i}=\sum _{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in \mathbb {Q} .$ Here $e_{1},\ldots ,e_{n}$ is a fixed basis for $K {\displaystyle K}$ , viewed as a $\mathbb {Q}$ -vector space. The rational numbers $a_{ij}$ are uniquely determined by $x {\displaystyle x}$ and the choice of a basis since any element of $K {\displaystyle K}$ can be uniquely represented as alinear combination of the basis elements. This way of associating a matrix to any element of the field $K {\displaystyle K}$ is called theregular representation. The square matrix $A {\displaystyle A}$ represents the effect of multiplication by $x {\displaystyle x}$ in the given basis. It follows that if the element $y {\displaystyle y}$ of $K {\displaystyle K}$ is represented by a matrix $B {\displaystyle B}$ , then the product $x y {\displaystyle xy}$ is represented by thematrix product $B A {\displaystyle BA}$ .Invariants of matrices, such as thetrace,determinant, andcharacteristic polynomial, depend solely on the field element $x {\displaystyle x}$ and not on the basis. In particular, the trace of the matrix $A(x)$ is called thetrace of the field element $x {\displaystyle x}$ and denoted ${\text{Tr}}(x)$ , and the determinant is called thenorm ofx and denoted $N(x)$ .

Now this can be generalized slightly by instead considering a field extension $K/L$ and giving an $L {\displaystyle L}$ -basis for $K {\displaystyle K}$ . Then, there is an associated matrix $A_{K/L}(x)$ , which has trace ${\text{Tr}}_{K/L}(x)$ and norm ${\text{N}}_{K/L}(x)$ defined as the trace and determinant of the matrix $A_{K/L}(x)$ .

Example

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Consider the field extension $\mathbb {Q} (\theta )$ with $\theta =\zeta _{3}{\sqrt[{3}]{2}}$ , where $\zeta _{3}$ denotes the cube root of unity $\exp(2\pi i/3).$ Then, we have a $\mathbb {Q}$ -basis given by $\{1,\zeta _{3}{\sqrt[{3}]{2}},(\zeta _{3}{\sqrt[{3}]{2}})^{2}\}$ since any $x\in \mathbb {Q} (\theta )$ can be expressed as some $\mathbb {Q}$ -linear combination: $x=a+b\zeta _{3}{\sqrt[{3}]{2}}+c(\zeta _{3}{\sqrt[{3}]{2}})^{2}=a+b\theta +c\theta ^{2}.$ We proceed to calculate the trace $T(x)$ and norm $N(x)$ of this number. To this end, we take an arbitrary $y\in \mathbb {Q} (\theta )$ where $y=y_{0}+y_{1}\theta +y_{2}\theta ^{2}$ and compute the product $x y {\displaystyle xy}$ . Writing this out gives ${\begin{aligned}xy=a(y_{0}+y_{1}\theta +y_{2}\theta ^{2})+\\b(2y_{2}+y_{0}\theta +y_{1}\theta ^{2})+\\c(2y_{1}+2y_{2}\theta +y_{0}\theta ^{2}).\end{aligned}}$ We can find the matrix $A(x)$ such that $xy=A(x)y$ by writing out the associated matrix equation giving ${\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}{\begin{bmatrix}y_{0}\\y_{1}\\y_{2}\end{bmatrix}}={\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\by_{0}+ay_{1}+2cy_{2}\\cy_{0}+by_{1}+ay_{2}\end{bmatrix}}$ showing that $A(x)={\begin{bmatrix}a&2c&2b\\b&a&2c\\c&b&a\end{bmatrix}}$ is the matrix that governs multiplication by the number $x {\displaystyle x}$ .

We can now easily compute the trace and determinant: $T(x)=3a$ , and $N(x)=a^{3}+2b^{3}+4c^{3}-6abc$ .

Properties

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By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is alinear function ofx, as expressed byTr(x +y) = Tr(x) + Tr(y),Tr(λx) =λ Tr(x), and the norm is a multiplicativehomogeneous function of degreen:N(xy) = N(x) N(y),N(λx) =λⁿ N(x). Hereλ is a rational number, andx,y are any two elements of $K {\displaystyle K}$ .

Thetrace form derived is abilinear form defined by means of the trace, as $Tr_{K/L}:K\otimes _{L}K\to L$ by $Tr_{K/L}(x\otimes y)=Tr_{K/L}(x\cdot y)$ and extending linearly. Theintegral trace form, an integer-valuedsymmetric matrix is defined as $t_{ij}={\text{Tr}}_{K/\mathbb {Q} }(b_{i}b_{j})$ , whereb₁, ...,b_n is an integral basis for $K {\displaystyle K}$ . Thediscriminant of $K {\displaystyle K}$ is defined as det(t). It is an integer, and is an invariant property of the field $K {\displaystyle K}$ , not depending on the choice of integral basis.

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

Example with integral basis

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Consider $K=\mathbb {Q} (x)$ , wherex satisfiesx³ − 11x² +x + 1 = 0. Then an integral basis is [1,x, 1/2(x² + 1)], and the corresponding integral trace form is ${\begin{bmatrix}3&11&61\\11&119&653\\61&653&3589\end{bmatrix}}.$

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 $K {\displaystyle K}$ on $K {\displaystyle K}$ . This basis element induces the identity map on the 3-dimensional vector space, $K {\displaystyle K}$ . The trace of the matrix of the identity map on a 3-dimensional vector space is 3.

The determinant of this is1304 = 2³·163, the field discriminant; in comparison theroot discriminant, or discriminant of the polynomial, is5216 = 2⁵·163.

Places

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Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.^[4]^[5] This situation changed with the discovery ofp-adic numbers byHensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field $K {\displaystyle K}$ into its various topologicalcompletions $K_{\mathfrak {p}}$ at once.

Aplace of a number field $K {\displaystyle K}$ is an equivalence class ofabsolute values on $K {\displaystyle K}$ ^[6]^{pg 9}. Essentially, an absolute value is a notion to measure the size of elements $x {\displaystyle x}$ of $K {\displaystyle K}$ . 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 $1 {\displaystyle 1}$ on all non-zero $x\in K$ . The second and third classes areArchimedean places andnon-Archimedean (or ultrametric) places. The completion of $K {\displaystyle K}$ with respect to a place $|\cdot |_{\mathfrak {p}}$ is given in both cases by takingCauchy sequences in $K {\displaystyle K}$ and dividing outnull sequences, that is, sequences $\{x_{n}\}_{n\in \mathbb {N} }$ such that $|x_{n}|_{\mathfrak {p}}\to 0$ tends to zero when $n {\displaystyle n}$ tends to infinity. This can be shown to be a field again, the so-called completion of $K {\displaystyle K}$ at the given place $|\cdot |_{\mathfrak {p}}$ , denoted $K_{\mathfrak {p}}$ .

For $K=\mathbb {Q}$ , the following non-trivial norms occur (Ostrowski's theorem): the (usual)absolute value, sometimes denoted $|\cdot |_{\infty }$ , which gives rise to the completetopological field of the real numbers $\mathbb {R}$ . On the other hand, for any prime number $p {\displaystyle p}$ , thep-adic absolute value is defined by

|q|_p =p⁻ⁿ, whereq =pⁿa/b anda andb are integers not divisible byp.

It is used to construct the $p {\displaystyle p}$ -adic numbers $\mathbb {Q} _{p}$ . In contrast to the usual absolute value, thep-adic absolute value getssmaller whenq is multiplied byp, leading to quite different behavior of $\mathbb {Q} _{p}$ as compared to $\mathbb {R}$ .

Note the general situation typically considered is taking a number field $K {\displaystyle K}$ and considering aprime ideal ${\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})$ for its associatedring of algebraic numbers ${\mathcal {O}}_{K}$ . 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 $|\cdot |_{\sigma }:K\to \mathbb {R} _{\geq 0}$ . This statement is a theorem also calledOstrowski's theorem.

Examples

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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^[6]^{pg 15-16}.

Archimedean places

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

Calculating the archimedean places of a number field $K {\displaystyle K}$ is done as follows: let $x {\displaystyle x}$ be a primitive element of $K {\displaystyle K}$ , with minimal polynomial $f {\displaystyle f}$ (over $\mathbb {Q}$ ). Over $\mathbb {R}$ , $f {\displaystyle f}$ 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 $r {\displaystyle r}$ of factors of degree one are necessarily real, and replacing $x {\displaystyle x}$ by $r {\displaystyle r}$ gives an embedding of $K {\displaystyle K}$ into $\mathbb {R}$ ; the number of such embeddings is equal to the number of real roots of $f {\displaystyle f}$ . Restricting the standard absolute value on $\mathbb {R}$ to $K {\displaystyle K}$ gives an archimedean absolute value on $K {\displaystyle K}$ ; such an absolute value is also referred to as areal place of $K {\displaystyle K}$ . On the other hand, the roots of factors of degree two are pairs ofconjugate complex numbers, which allows for two conjugate embeddings into $\mathbb {C}$ . Either one of this pair of embeddings can be used to define an absolute value on $K {\displaystyle K}$ , which is the same for both embeddings since they are conjugate. This absolute value is called acomplex place of $K {\displaystyle K}$ .^[7]^[8]

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

Non-Archimedean or ultrametric places

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To find the non-Archimedean places, let again $f {\displaystyle f}$ and $x {\displaystyle x}$ be as above. In $\mathbb {Q} _{p}$ , $f {\displaystyle f}$ 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 $f_{i}$ , we may suppose that $x {\displaystyle x}$ satisfies $f_{i}$ and obtain an embedding of $K {\displaystyle K}$ into an algebraic extension of finite degree over $\mathbb {Q} _{p}$ . Such alocal 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 $\mathbb {Q} _{p}$ . By using this $p {\displaystyle p}$ -adic norm map $N_{f_{i}}$ for the place $f_{i}$ , we may define an absolute value corresponding to a given $p {\displaystyle p}$ -adically irreducible factor $f_{i}$ of degree $m {\displaystyle m}$ by $|y|_{f_{i}}=|N_{f_{i}}(y)|_{p}^{1/m}$ Such an absolute value is called anultrametric, non-Archimedean or $p {\displaystyle p}$ -adic place of $K {\displaystyle K}$ .

For any ultrametric placev we have that |x|_v ≤ 1 for anyx in ${\mathcal {O}}_{K}$ , since the minimal polynomial forx has integer factors, and hence itsp-adic factorization has factors inZ_p. Consequently, the norm term (constant term) for each factor is ap-adic integer, and one of these is the integer used for defining the absolute value forv.

Prime ideals inO_K

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For an ultrametric placev, the subset of ${\mathcal {O}}_{K}$ defined by |x|_v < 1 is anideal ${\mathfrak {p}}$ of ${\mathcal {O}}_{K}$ . This relies on the ultrametricity ofv: givenx andy in ${\mathfrak {p}}$ , then

|x +y|_v ≤ max (|x|_v, |y|_v) < 1.

Actually, ${\mathfrak {p}}$ is even aprime ideal.

Conversely, given a prime ideal ${\mathfrak {p}}$ of ${\mathcal {O}}_{K}$ , adiscrete valuation can be defined by setting $v_{\mathfrak {p}}(x)=n$ wheren is the biggest integer such that $x\in {\mathfrak {p}}^{n}$ , then-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 ${\mathcal {O}}_{K}$ . For $K=\mathbb {Q}$ , this gives back Ostrowski's theorem: any prime ideal inZ (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 oflocalizations of ${\mathcal {O}}_{K}$ . 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 |_v ≤ 1. By the ultrametric property $T {\displaystyle T}$ is a ring. Moreover, it contains ${\mathcal {O}}_{K}$ . For every elementx of $K {\displaystyle K}$ , at least one ofx orx⁻¹ is contained in $T {\displaystyle T}$ . Actually, sinceK^×/T^× can be shown to be isomorphic to the integers, $T {\displaystyle T}$ is adiscrete valuation ring, in particular alocal ring. Actually, $T {\displaystyle T}$ is just the localization of ${\mathcal {O}}_{K}$ at the prime ideal ${\mathfrak {p}}$ , so $T={\mathcal {O}}_{K,{\mathfrak {p}}}$ . Conversely, ${\mathfrak {p}}$ 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.

Lying over theorem and places

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Some of the basic theorems in algebraic number theory are thegoing 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 $L/K$ . We say that an ideal ${\mathfrak {o}}\subset {\mathcal {O}}_{L}$ lies over ${\mathfrak {p}}$ if ${\mathfrak {o}}\cap {\mathcal {O}}_{K}={\mathfrak {p}}$ . Then, one incarnation of the theorem states a prime ideal in ${\text{Spec}}({\mathcal {O}}_{L})$ lies over ${\mathfrak {p}}$ , hence there is always a surjective map ${\text{Spec}}({\mathcal {O}}_{L})\to {\text{Spec}}({\mathcal {O}}_{K})$ induced from the inclusion ${\mathcal {O}}_{K}\hookrightarrow {\mathcal {O}}_{L}$ . 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 $p {\displaystyle p}$ is a place of $K {\displaystyle K}$ , then there are places $v {\displaystyle v}$ of $L {\displaystyle L}$ that divide $p {\displaystyle p}$ , in the sense that their induced prime ideals divide the induced prime ideal of $p {\displaystyle p}$ in ${\text{Spec}}({\mathcal {O}}_{L})$ .In fact, this observation is useful^[6]^{pg 13} while looking at the base change of an algebraic field extension of $\mathbb {Q}$ to one of its completions $\mathbb {Q} _{p}$ . If we write $K={\frac {\mathbb {Q} [X]}{Q(X)}}$ and write $\theta$ for the induced element of $X\in K$ , we get a decomposition of $K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}$ . Explicitly, this decomposition is ${\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&={\frac {\mathbb {Q} [X]}{Q(X)}}\otimes _{\mathbb {Q} }\mathbb {Q} _{p}\\&={\frac {\mathbb {Q} _{p}[X]}{Q(X)}}\end{aligned}}$ furthermore, the induced polynomial $Q(X)\in \mathbb {Q} _{p}[X]$ decomposes as $Q(X)=\prod _{v|p}Q_{v}$ because ofHensel's lemma^[11]^{pg 129-131}; hence ${\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&\cong {\frac {\mathbb {Q} _{p}[X]}{\prod _{v|p}Q_{v}(X)}}\\&\cong \bigoplus _{v|p}K_{v}\end{aligned}}$ Moreover, there are embeddings ${\begin{aligned}i_{v}:&K\to K_{v}\\&\theta \mapsto \theta _{v}\end{aligned}}$ where $\theta _{v}$ is a root of $Q_{v}$ giving $K_{v}=\mathbb {Q} _{p}(\theta _{v})$ ; hence we could write $K_{v}=i_{v}(K)\mathbb {Q} _{p}$ as subsets of $\mathbb {C} _{p}$ (which is the completion of the algebraic closure of $\mathbb {Q} _{p}$ ).

Ramification

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Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps $f:X\to Y$ such that thepreimages of all pointsy inY consist only of finitely many points): the cardinality of thefibersf⁻¹(y) will generally have the same number of points, but it occurs that, in special pointsy, this number drops. For example, the map

\mathbb {C} \to \mathbb {C} ,z\mapsto z^{n}

hasn points in each fiber overt, namely then (complex) roots oft, except in t =0, where the fiber consists of only one element,z = 0. One says that the map is "ramified" in zero. This is an example of abranched covering ofRiemann surfaces. This intuition also serves to defineramification in algebraic number theory. Given a (necessarily finite) extension of number fields $K/L$ , a prime idealp of ${\mathcal {O}}_{L}$ generates the idealpO_K of ${\mathcal {O}}_{K}$ . This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by

pO_{$K {\displaystyle K}$} =q₁^e₁q₂^e₂ ⋯q_m^e_m

with uniquely determined prime idealsq_i of ${\mathcal {O}}_{K}$ and numbers (called ramification indices)e_i. Whenever one ramification index is bigger than one, the primep is said to ramify in $K {\displaystyle K}$ .

The connection between this definition and the geometric situation is delivered by the map ofspectra of rings $\mathrm {Spec} {\mathcal {O}}_{K}\to \mathrm {Spec} {\mathcal {O}}_{L}$ . In fact,unramified morphisms ofschemes inalgebraic geometry are a direct generalization of unramified extensions of number fields.

Ramification is a purely local property, i.e., depends only on the completions around the primesp andq_i. Theinertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.

An example

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

f(x) =x³ −x − 1 = 0,

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

f(x) = (x + 181)(x² − 181x − 38) =gh.

Substitutingx =y + 10 in the first factorg modulo 529 yieldsy + 191, so the valuation | y |_g fory given byg is | −191 |₂₃ = 1. On the other hand, the same substitution inh yieldsy² − 161y − 161 modulo 529. Since 161 = 7 × 23,

\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 factorh 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 $K {\displaystyle K}$ can be computed in this way usingresultants. If, for exampley =x² −x − 1, using the resultant to eliminatex between this relationship andf =x³ −x − 1 = 0 givesy³ − 5y² + 4y − 1 = 0. If instead we eliminate with respect to the factorsg andh off, we obtain the corresponding factors for the polynomial fory, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations ofy forg andh (which are both 1 in this instance.)

Dedekind discriminant theorem

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Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in $\mathbb {Q} _{p}$ wherep divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a primep divides the discriminant, then there is ap-place that ramifies. For this converse the field discriminant is needed. This is theDedekind discriminant theorem. In the example above, the discriminant of the number field $\mathbb {Q} (x)$ withx³ − 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 $K {\displaystyle K}$ .

Galois groups and Galois cohomology

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Generally in abstract algebra, field extensionsK /L can be studied by examining theGalois group Gal(K /L), consisting of field automorphisms of $K {\displaystyle K}$ leaving $L {\displaystyle L}$ elementwise fixed. As an example, the Galois group $\mathrm {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )$ of the cyclotomic field extension of degreen (see above) is given by (Z/nZ)^×, the group of invertible elements inZ/nZ. This is the first stepstone intoIwasawa theory.

In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extensionK /K of thealgebraic closure, leading to theabsolute Galois groupG := Gal(K /K) or just Gal(K), and to the extension $K/\mathbb {Q}$ . Thefundamental theorem of Galois theory links fields in between $K {\displaystyle K}$ and its algebraic closure and closed subgroups of Gal(K). For example, theabelianization (the biggest abelian quotient)G^ab ofG corresponds to a field referred to as the maximalabelian extensionK^ab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By theKronecker–Weber theorem, the maximal abelian extension of $\mathbb {Q}$ is the extension generated by allroots of unity. For more general number fields,class field theory, specifically theArtin reciprocity law gives an answer by describingG^ab in terms of theidele class group. Also notable is theHilbert class field, the maximal abelian unramified field extension of $K {\displaystyle K}$ . It can be shown to be finite over $K {\displaystyle K}$ , its Galois group over $K {\displaystyle K}$ is isomorphic to the class group of $K {\displaystyle K}$ , in particular its degree equals the class numberh of $K {\displaystyle K}$ (see above).

In certain situations, the Galois groupacts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use ofgroup cohomology for the Galois group Gal(K), also known asGalois cohomology, which in the first place measures the failure of exactness of taking Gal(K)-invariants, but offers deeper insights (and questions) as well. For example, the Galois groupG of a field extensionL /K acts onL^×, the nonzero elements ofL. This Galois module plays a significant role in many arithmeticdualities, such asPoitou-Tate duality. TheBrauer group of $K {\displaystyle K}$ , originally conceived to classifydivision algebras over $K {\displaystyle K}$ , can be recast as a cohomology group, namely H²(Gal (K,K^×)).

Local-global principle

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Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion ofsheaves reifies that idea intopology andgeometry.

Local and global fields

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Number fields share a great deal of similarity with another class of fields much used inalgebraic geometry known asfunction fields ofalgebraic curves overfinite fields. An example isK_p(T). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are thecoordinate rings (the quotient fields of which are the function fields in question) of curves. Therefore, both types of field are calledglobal 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 correspondinglocal fields. For number fields $K {\displaystyle K}$ , the local fields are the completions of $K {\displaystyle K}$ at all places, including the archimedean ones (seelocal 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

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Main article:Hasse principle

A prototypical question, posed at a global level, is whether some polynomial equation has a solution in $K {\displaystyle K}$ . If this is the case, this solution is also a solution in all completions. Thelocal-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 $K {\displaystyle K}$ , which is often easier, since analytic methods (classical analytic tools such asintermediate value theorem at the archimedean places andp-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, wherelocal class field theory is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completionsK_v can be explicitly determined, whereas the Galois groups of global fields, even of $\mathbb {Q}$ are far less understood.

Adeles and ideles

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

Notes

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^Ireland, Kenneth;Rosen, Michael (1998),A Classical Introduction to Modern Number Theory, Berlin, New York:Springer-Verlag,ISBN 978-0-387-97329-6, Ch. 1.4
^Bloch, Spencer;Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives",The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Boston, MA: Birkhäuser Boston, pp. 333–400,MR 1086888
^Narkiewicz 2004, §2.2.6
^Kleiner, Israel (1999), "Field theory: from equations to axiomatization. I",The American Mathematical Monthly,106 (7):677–684,doi:10.2307/2589500,JSTOR 2589500,MR 1720431,To Dedekind, then, fields were subsets of the complex numbers.
^Mac Lane, Saunders (1981), "Mathematical models: a sketch for the philosophy of mathematics",The American Mathematical Monthly,88 (7):462–472,doi:10.2307/2321751,JSTOR 2321751,MR 0628015,Empiricism sprang from the 19th-century view of mathematics as almost coterminal with theoretical physics.
^^a ^b ^cGras, Georges (2003).Class field theory : from theory to practice. Berlin.ISBN 978-3-662-11323-3.OCLC 883382066.{{cite book}}: CS1 maint: location missing publisher (link)
^Cohn, Chapter 11 §C p. 108
^Conrad
^Cohn, Chapter 11 §C p. 108
^Conrad
^Neukirch, Jürgen (1999).Algebraic Number Theory. Berlin, Heidelberg: Springer Berlin Heidelberg.ISBN 978-3-662-03983-0.OCLC 851391469.

References

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Keune, Frans (2023).Number Fields. Radboud University Press.ISBN 9789493296039.
Keith Conrad,http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf
Janusz, Gerald J. (1996),Algebraic Number Fields (2nd ed.), Providence, R.I.:American Mathematical Society,ISBN 978-0-8218-0429-2
Helmut Hasse,Number Theory, SpringerClassics in Mathematics Series (2002)
Serge Lang,Algebraic Number Theory, second edition, Springer, 2000
Richard A. Mollin,Algebraic Number Theory, CRC, 1999
Ram Murty,Problems in Algebraic Number Theory, Second Edition, Springer, 2005
Narkiewicz, Władysław (2004),Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin:Springer-Verlag,ISBN 978-3-540-21902-6,MR 2078267
Neukirch, Jürgen (1999),Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Berlin, New York:Springer-Verlag,ISBN 978-3-540-65399-8,MR 1697859,Zbl 0956.11021
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000),Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin, New York:Springer-Verlag,ISBN 978-3-540-66671-4,MR 1737196,Zbl 1136.11001
André Weil,Basic Number Theory, third edition, Springer, 1995

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Movatterモバイル変換

Algebraic number field

Definition

Prerequisites

Definition

Examples

Non-examples

Algebraicity, and ring of integers

Unique factorization

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

Bases for number fields

Integral basis

Power basis

Regular representation, trace and discriminant

Example

Properties

Example with integral basis

Places

Examples

Archimedean places

Non-Archimedean or ultrametric places

Prime ideals inO_K

Lying over theorem and places

Ramification

An example

Dedekind discriminant theorem

Galois groups and Galois cohomology

Local-global principle

Local and global fields

Hasse principle

Adeles and ideles

See also

Generalizations

Algebraic number theory

Class field theory

Notes

References

Movatterモバイル変換

Definition

Prerequisites

Definition

Examples

Non-examples

Algebraicity, and ring of integers

Unique factorization

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

Bases for number fields

Integral basis

Power basis

Regular representation, trace and discriminant

Example

Properties

Example with integral basis

Places

Examples

Archimedean places

Non-Archimedean or ultrametric places

Prime ideals inOK

Lying over theorem and places

Ramification

An example

Dedekind discriminant theorem

Galois groups and Galois cohomology

Local-global principle

Local and global fields

Hasse principle

Adeles and ideles

See also

Generalizations

Algebraic number theory

Class field theory

Notes

References

Prime ideals inO_K