Full body

In the mathematical subfield of algebra , a complete body (also a complete weighted body ) is a weighted body , which is a complete space with the metric resulting from the weighting .

Standard example for a complete body is for an incomplete body . In these two bodies, the absolute amount provides the evaluation. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$

For ordered bodies , one has a second term of completeness in addition to order completeness (metric completeness or Cauchy completeness), but for Archimedean bodies (such as or ) the two are equivalent: an ordered body is Archimedean and Cauchy-complete if and only if it is orderly . However, there are unordered solids (like or ) that are metrically complete. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {C} }$

Explanations

A rated body is a body with a rating, i. H. a mapping into the real numbers ${\displaystyle (K,+,\cdot )}$

${\displaystyle \phi \colon K\to \mathbb {R} _{\geq 0}}$,

which the conditions

${\displaystyle \phi (a)>0}$for , and ${\displaystyle a\not =0}$ ${\displaystyle \phi (0)=0}$

${\displaystyle \phi (a\cdot b)=\phi (a)\cdot _{\mathbb {R} }\phi (b)}$ for all ${\displaystyle a,b}$

${\displaystyle \phi (a+b)\leq \phi (a)+_{\mathbb {R} }\phi (b)}$ for all ${\displaystyle a,b}$

Fulfills.

The rating induces a metric on through ${\displaystyle d}$ ${\displaystyle K}$

${\displaystyle d(x,y):=\phi (x-y)}$.

A weighted field is called complete if every Cauchy sequence converges in with the induced metric . ${\displaystyle K}$

Generalizations

The term "whole body" suggests not only looking at valued bodies, but more generally, metric bodies. The nLab defines a complete body as a complete space and additionally demands the continuity of the body operations, i.e. the images

1. ${\displaystyle +\colon K\times K\to K}$
2. ${\displaystyle \cdot \colon K\times K\to K}$

with respect to the topology generated by the metric are continuous . This continuity follows automatically from the above-mentioned properties of an evaluation.

Examples

• The body of the real numbers with the metric . ${\displaystyle \mathbb {R} }$ ${\displaystyle d(x,y)=|x-y|}$
• The body of complex numbers with the metric . ${\displaystyle \mathbb {C} }$ ${\displaystyle d(x,y)=|x-y|}$
• The field of the p-adic numbers with the metric defined by the p-adic norm . ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle |.|_{p}}$ ${\displaystyle d(x,y)=|x-y|_{p}}$
• The oblique body of the quaternions with the metric . ${\displaystyle \mathbb {H} }$ ${\displaystyle d(x,y)=|x-y|}$

Completion of rated bodies

definition

Let be a valued body and the metric induced by the valuation. The completion of this metric is a complete body, which is denoted by. ${\displaystyle K}$ ${\displaystyle d}$ ${\displaystyle K}$ ${\displaystyle {\overline {K}}}$

Examples

• Starting from the field of rational numbers with the p-adic valuation , you get the field of p-adic numbers as a completion . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} _{p}}$
• Starting from the field of rational numbers with the absolute amount, you get the field of real numbers as a completion.
• Starting from the rational function body and the evaluation given by the zero order at the zero point, one obtains the body of the formal Laurent series as a completion . ${\displaystyle \mathbb {Q} (x)}$ ${\displaystyle \mathbb {Q} ((x))}$

Individual evidence

1. ^ Lexicon of Mathematics . tape 5 . Spektrum Akademischer Verlag, Heidelberg 2002, ISBN 3-8274-0437-1 , complete body, p. 352 ( Spektrum.de [accessed on March 26, 2019]).
2. ↑ Toby Bartels, et al. : complete space. In: nLab. July 12, 2018, accessed March 26, 2019 . 

Web links

• Ramanathan: Lectures on the algebraic theory of fields (Chapter 8.4)