# nLab rational number

Rational numbers

### Context

#### Algebra

higher algebra

universal algebra

# Rational numbers

## Definition

A rational number is a fraction of two integer numbers.

### As a field

The field of rational numbers, $\mathbb{Q}$, is the field of fractions of the commutative ring of integers, $\mathbb{Z}$, hence the field consisting of formal fractions (“ratios”) of integers.

### As a commutative ring

Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers. The positive integers are embedded into every commutative ring $R$: there is an injection $inj:\mathbb{N}^+\to\R$ such that $inj(1) = 1$ and $inj(s(n)) = inj(n) + 1$ for all $n:\mathbb{N}^+$.

Suppose $R$ has an injection $inv:\mathbb{N}^+\to\R$ such that $inj(n) \cdot inv(n) = 1$ and $\inv(n) \cdot \inj(n) = 1$ for all $n:\mathbb{N}^+$. Then $R$ is called a $\mathbb{Q}$-algebra, and the commutative ring of rational numbers $\mathbb{Q}$ is the initial commutative $\mathbb{Q}$-algebra.

It can then be proven from the ring axioms and the properties of the integers that every rational number apart from zero and has a multiplicative inverse, making $\mathbb{Q}$ a field.

### As an directed colimit in CRing

Let $(\mathbb{N},\leq)$ be the directed set of positive integers, and let $A:\mathbb{N}\to CRing$ be a family of commutative rings where $A_n$ is defined to be $\mathbb{Z}[1/n!]$, the localization of the integers $\mathbb{Z}$ away from the factorial $n!$, and for $i, j:\mathbb{N}$, $i\leq j$, there is a commutative ring homomorphism from $f_{ij}:\mathbb{Z}[1/i!]\to\mathbb{Z}[1/j!]$, with $f_{ii}$ being the identity commutative ring homomorphism on $\mathbb{Z}[1/i!]$. Then the commutative ring of rational numbers $\mathbb{Q}$ is the directed colimit $\underset{\to}\lim_i A_i$ of the system.

### As an abelian group

Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers and let $(\mathbb{Z},0,+,-,1)$ be the free abelian group on the set ${1}$.

Let $A$ be an abelian group containing $\mathbb{Z}$ as an abelian subgroup. The positive integers are embedded into the function abelian group $A \to A$, with $id_A:A \to A$ being the identity function on $A$; i.e. there is an injection $inj:\mathbb{N}^+\to (A \to A)$ such that $inj(1) = id_A$ and $inj(s(n)) = inj(n) + id_A$ for all $n:\mathbb{N}^+$.

Suppose $A$ has an injection $inv:\mathbb{N}^+\to (A \to A)$ such that for all $n:\mathbb{N}^+$, $inj(n) \circ inv(n) = id_A$ and $inv(n) \circ inj(n) = id_A$. Then the abelian group of rational numbers $\mathbb{Q}$ is the initial such abelian group.

## Properties

### Algebraic closure

The algebraic closure $\overline{\mathbb{Q}}$ of the rational numbers is called the field of algebraic numbers. The absolute Galois group $Gal(\overline{\mathbb{Q}}\vert \mathbb{Q})$ has some curious properties, see there.

### Topologies

There are several interesting topologies on $\mathbb{Q}$ that make $\mathbb{Q}$ into a topological group under addition, allowing us to define interesting fields by taking the completion with respect to this topology:

1. The discrete topology is the most obvious, which is already complete.

2. The absolute-value topology is defined by the metric $d(x,y) \coloneqq {|x - y|}$; the completion is the field of real numbers.

(This topology is totally disconnected (this exmpl.))

3. Fixing a prime number $p$, the $p$-adic topology is defined by the ultrametric $d(x,y) \coloneqq 1/n$ where $n$ is the highest exponent on $p$ in the prime factorization of ${|x - y|}$; the completion is the field of $p$-adic numbers.

According to Ostrowski's theorem this are the only possibilities.

Interestingly, (2) cannot be interpreted as a localic group, although the completion $\mathbb{R}$ can. (Probably the same holds for (3); I need to check.)

Last revised on May 8, 2021 at 14:34:29. See the history of this page for a list of all contributions to it.