Normirana polja

magistrsko delo

Špela Čebul (Author), Mateja Grašič (Mentor)

Abstract

Vsaka norma na kolobarju na običajen način porodi metriko. Posledično lahko definiramo pojem poln metrični prostor in preko njega poln kolobar. Dokažemo, da lahko poljuben kolobar K vložimo v poln kolobar K' na tak način, da je K gosti v K'. Izrek Ostrovskega pove, da je vsaka netrivialna norma na polju racionalnih števil Q ekvivalentna bodisi standardni normi, pridobljeni iz absolutne vrednosti, bodisi p-adični normi, kjer je p praštevilo. Ker za ekvivalentne norme na nekem polju dokažemo, da inducirajo enako topologijo na tem polju, potem na polju racionalnih števil obstajata natanko dve različni napolnitvi. Napolnitev polja racionalnih števil glede na standardno normo je polje realnih števil R, glede na p-adično normo pa polje p-adičnih števil Q_p.

Keywords

magistrska dela;kolobarji;polja;faktorski kolobarji;norme;popolni kolobarji;napolnitev;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[Š. Čebul]
UDC:	512.55(043.2)
COBISS:	23834632
Views:	738
Downloads:	58
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Normed fields
Secondary abstract:	Every norm on the ring generates a metrics on a usual way. Consequently we can define a complete metric space and through it a complete ring. We prove, that we can imbed a ring K into the complete ring K' in such a way, that K is dense in K'. The Ostrovski's theorem states, that any non-trivial norm of the field Q of rational numbers is equivalent to the usual norm which is obtained by means of the standard absolute value, or to a p-adic norm, where p is a prime number. As we prove for the equivalent norms on a field, that they induce the same topology on this field, there are exactly two different completions of the field of rational numbers. The completion of the field of rational numbers with respect to the standard norm is a field of real numbers and with respect to the p-adic norm a field of p-adic numbers.
Secondary keywords:	master theses;rings;fields;factorial rings;normas;complete rings;completion;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	IX, 67 f.
ID:	10901608