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Title:Normirana polja
Authors:ID Čebul, Špela (Author)
ID Grašič, Mateja (Mentor) More about this mentor... New window
Files:.pdf MAG_Cebul_Spela_2018.pdf (511,58 KB)
MD5: 07D0DF816F5C314E017B1AA2E26D8344
PID: 20.500.12556/dkum/62b35c9c-89f1-4f70-8658-4088d0add331
 
Language:Slovenian
Work type:Master's thesis/paper
Typology:2.09 - Master's Thesis
Organization:FNM - Faculty of Natural Sciences and Mathematics
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:Kolobar, polje, faktorski kolobar, norma, poln kolobar, napolnitev.
Place of publishing:Maribor
Publisher:[Š. Čebul]
Year of publishing:2018
PID:20.500.12556/DKUM-69496 New window
UDC:512.55(043.2)
COBISS.SI-ID:23834632 New window
NUK URN:URN:SI:UM:DK:XJUQVNYB
Publication date in DKUM:29.05.2018
Views:1426
Downloads:84
Metadata:XML DC-XML DC-RDF
Categories:FNM
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Licences

License:CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
Link:http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:30.01.2018

Secondary language

Language:English
Title:Normed fields
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.
Keywords:Ring, field, factorial ring, norm, complete ring, completion.


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