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 $F$ 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:

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

Complete metric

A metric on a set $F$ is a function $d:F^2\to [0,\mathrm{\infty })$, 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 $x,y,z\in F$:

1. $d(x,y)=0$ if and only if $x=y$
2. $d(x,y)=d(y,x)$
3. $d(x,y)\le d(x,z)+d(z,y)$

A sequence $x_n$ in the space is Cauchy with respect to this metric if for all $ϵ>0$ there exists an $N\in \mathbb{N}$ such that for all $n,m\ge N$ we have $d(x_n,x_m)<ϵ$, 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 $ϵ>0$ there exists an $N\in \mathbb{N}$ such that for all $n\ge N$ we have $d(x_n,x)<ϵ$. 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 $ℂ$, 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 ${ℂ}_p.$

See also

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