Complete field

In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. A field supports the elementary operations of addition, subtraction, multiplication, and division, while a metric represents the distance between two points in the set. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).

Definitions

Field

A field is a set ${\displaystyle F}$ with binary operations ${\displaystyle +}$ and ${\displaystyle \cdot }$ (called addition and multiplication, respectively), along with elements ${\displaystyle 0}$ and ${\displaystyle 1}$ such that for all ${\displaystyle a,b,c\in F}$, the following relations hold:^[1]

1. ${\displaystyle a+(b+c)=(a+b)+c}$
2. ${\displaystyle a+b=b+a}$
3. ${\displaystyle a+0=a=0+a}$
4. ${\displaystyle a+x=0}$ has a solution
5. ${\displaystyle a(bc)=(ab)c}$
6. ${\displaystyle ab=ba}$
7. ${\displaystyle a(b+c)=ab+ac}$ and ${\displaystyle (a+b)c=ac+bc}$
8. ${\displaystyle a1=a=1a}$
9. ${\displaystyle ax=1}$ has a solution for ${\displaystyle a\neq 0}$

Complete metric

A metric on a set ${\displaystyle F}$ is a function ${\displaystyle d:F^{2}\to [0,\infty )}$, that is, it takes two points in ${\displaystyle F}$ and sends them to a non-negative real number, such that the following relations hold for all ${\displaystyle x,y,z\in F}$:^[2]

1. ${\displaystyle d(x,y)=0}$ if and only if ${\displaystyle x=y}$
2. ${\displaystyle d(x,y)=d(y,x)}$
3. ${\displaystyle d(x,y)\leq d(x,z)+d(z,y)}$

A sequence ${\displaystyle x_{n}}$ in the space is Cauchy with respect to this metric if for all ${\displaystyle \epsilon >0}$ there exists an ${\displaystyle N\in \mathbb {N} }$ such that for all ${\displaystyle n,m\geq N}$ we have ${\displaystyle d(x_{n},x_{m})<\epsilon }$, and a metric is then complete if every Cauchy sequence in the metric space converges, that is, there is some ${\displaystyle x\in F}$ where for all ${\displaystyle \epsilon >0}$ there exists an ${\displaystyle N\in \mathbb {N} }$ such that for all ${\displaystyle n\geq N}$ we have ${\displaystyle d(x_{n},x)<\epsilon }$. Every convergent sequence is Cauchy, however the converse does not hold in general.^[2][3]

Constructions

Real and complex numbers

The real numbers are the field with the standard Euclidean metric ${\displaystyle |x-y|}$, and this measure is complete.^[2] Extending the reals by adding the imaginary number ${\displaystyle i}$ satisfying ${\displaystyle i^{2}=-1}$ gives the field ${\displaystyle \mathbb {C} }$, which is also a complete field.^[3]

p-adic

The p-adic numbers are constructed from ${\displaystyle \mathbb {Q} }$ by using the p-adic absolute value

${\displaystyle v_{p}(a/b)=v_{p}(a)-v_{p}(b)}$

where ${\displaystyle a,b\in \mathbb {Z} .}$ Then using the factorization ${\displaystyle a=p^{n}c}$ where ${\displaystyle p}$ does not divide ${\displaystyle c,}$ its valuation is the integer ${\displaystyle n}$. The completion of ${\displaystyle \mathbb {Q} }$ by ${\displaystyle v_{p}}$ is the complete field ${\displaystyle \mathbb {Q} _{p}}$ called the p-adic numbers. This is a case where the field is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted ${\displaystyle \mathbb {C} _{p}.}$^[4]