Complete field

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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 $F$ with binary operations $+$ and $\cdot$ (called addition and multiplication, respectively), along with elements $0$ and $1$ such that for all $a,b,c\in F$ , the following relations hold:

$a+(b+c)=(a+b)+c$
$a+b=b+a$
$a+0=a=0+a$
$a+x=0$ has a solution
$a(bc)=(ab)c$
$ab=ba$
$a(b+c)=ab+ac$ and $(a+b)c=ac+bc$
$a1=a=1a$
$ax=1$ has a solution for $a\neq 0$

Complete metric

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

$d(x,y)=0$ if and only if $x=y$
$d(x,y)=d(y,x)$
$d(x,y)\leq d(x,z)+d(z,y)$

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

Constructions

Real and complex numbers

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

p-adic

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

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

where $a,b\in \mathbb {Z}.$ Then using the factorization $a=p^{n}c$ where $p$ does not divide $c,$ its valuation is the integer $n$ . The completion of $\mathbb {Q}$ by $v_{p}$ is the complete field $\mathbb {Q} _{p}$ 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 $\mathbb {C} _{p}.$