# nLab number field

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A number field is a finite field extension of the field of rational numbers, $\mathbb{Q}$, In other words, a field $k$ of characteristic zero such that under the field homomorphism $i: \mathbb{Q} \hookrightarrow k$, the field $k$ is a finite-dimensional vector space over $\mathbb{Q}$ with respect to the scalar multiplication action of $\mathbb{Q}$

$\mathbb{Q} \otimes k \stackrel{i \otimes 1}{\to} k \otimes k \stackrel{mult}{\to} k$

on the underlying additive group of $k$.

## Examples

• the rational numbers $\mathbb{Q}$;

• the Gaussian numbers $\mathbb{Q}(i)$;

• for certain $d$ the quadratic field? $\mathbb{Q}(\sqrt{d})$;

• the cyclotomic fields $\mathbb{Q}(\zeta_n)$

Counterexamples:

• the real numbers $\mathbb{R}$ and complex numbers $\mathbb{C}$ are not number fields, since, while being vector spaces over $\mathbb{Q}$, they are not finite dimensional as $\mathbb{Q}$-vector spaces.

## Applications

Number fields are the basic objects of study in algebraic number theory. For example, one is typically interested in the arithmetic structure of $k$, including for example the structure of the ring of algebraic integers $\mathcal{O}_k$ in $k$, the decomposition of primes in $\mathbb{Z}$ in terms of prime ideals in $\mathcal{O}_k$, the structure of the unit group of $\mathcal{O}_k$, the structure of the ideal class group, the detailed study of the zeta function of $k$, and much more.

## Properties

### As global fields

Number fields $k$ are examples of global fields, in fact they are the global fields of characteristic zero. They are often studied in terms of how they embed in their rings of adeles $\mathbb{A}_k$, which are built from the local completions of $k$.

### Class number formula

Given a number field $K$, the Dedekind zeta function $\zeta_K$ of $K$ has a simple pole at $s = 1$. The class number formula says that its residue there is proportional the product of the regulator with the class number of $K$

$\underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.$

### Function field analogy

function field analogy

number fields (“function fields of curves over F1”)function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
$\mathbb{Z}$ (integers)$\mathbb{F}_q[z]$ (polynomials, function algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
$\mathbb{Q}$ (rational numbers)$\mathbb{F}_q(z)$ (rational functions)meromorphic functions on complex plane
$p$ (prime number/non-archimedean place)$x \in \mathbb{F}_p$$x \in \mathbb{C}$
$\infty$ (place at infinity)$\infty$
$Spec(\mathbb{Z})$ (Spec(Z))$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)complex plane
$Spec(\mathbb{Z}) \cup place_{\infty}$$\mathbb{P}_{\mathbb{F}_q}$ (projective line)Riemann sphere
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)$\frac{\partial}{\partial z}$ (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
$\mathbb{Z}_p$ (p-adic integers)$\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$)$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)formal disks in $X$
$\mathbb{Q}_p$ (p-adic numbers)$\mathbb{F}_q((z-x))$ (Laurent series around $x$)$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
$\mathcal{O}_K$ (ring of integers)$\mathcal{O}_{\Sigma}$ (structure sheaf)
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)$\Sigma$ (arithmetic curve)$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)$\frac{\partial}{\partial z}$
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
$v$ prime ideal in ring of integers $\mathcal{O}_K$$x \in \Sigma$$x \in \Sigma$
$K_v$ (formal completion at $v$)$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
$\mathcal{O}_{K_v}$ (ring of integers of formal completion)$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
$\mathbb{A}_K$ (ring of adeles)$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
$\mathcal{O}$$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
Galois group$\pi_1(\Sigma)$ fundamental group
Galois representationflat connection (“local system”) on $\Sigma$
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functionsHasse-Weil zeta function

Revised on August 25, 2014 23:55:55 by Urs Schreiber (89.204.138.168)