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rational number

Rational numbers

Context

Arithmetic

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 ( +,1: +,s: + +)(\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 RR: there is an injection inj: +Rinj:\mathbb{N}^+\to\R such that inj(1)=1inj(1) = 1 and inj(s(n))=inj(n)+1inj(s(n)) = inj(n) + 1 for all n: +n:\mathbb{N}^+.

Suppose RR has an injection inv: +Rinv:\mathbb{N}^+\to\R such that inj(n)inv(n)=1inj(n) \cdot inv(n) = 1 and inv(n)inj(n)=1\inv(n) \cdot \inj(n) = 1 for all n: +n:\mathbb{N}^+. Then RR 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:CRingA:\mathbb{N}\to CRing be a family of commutative rings where A nA_n is defined to be [1/n!]\mathbb{Z}[1/n!], the localization of the integers \mathbb{Z} away from the factorial n!n!, and for i,j:i, j:\mathbb{N}, iji\leq j, there is a commutative ring homomorphism from f ij:[1/i!][1/j!]f_{ij}:\mathbb{Z}[1/i!]\to\mathbb{Z}[1/j!], with f iif_{ii} being the identity commutative ring homomorphism on [1/i!]\mathbb{Z}[1/i!]. Then the commutative ring of rational numbers \mathbb{Q} is the directed colimit lim iA i\underset{\to}\lim_i A_i of the system.

As an abelian group

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

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

Suppose AA has an injection inv: +(AA)inv:\mathbb{N}^+\to (A \to A) such that for all n: +n:\mathbb{N}^+, inj(n)inv(n)=id Ainj(n) \circ inv(n) = id_A and inv(n)inj(n)=id Ainv(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(¯|)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)|xy|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 pp, the pp-adic topology is defined by the ultrametric d(x,y)1/nd(x,y) \coloneqq 1/n where nn is the highest exponent on pp in the prime factorization of |xy|{|x - y|}; the completion is the field of pp-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.