p-adične norme in p-adična števila

Abstract

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Language:	Slovenian
Year of publishing:	2014
Source:	Ljubljana
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[I. Femc]
UDC:	511(043.2)
COBISS:	10203977
Views:	851
Downloads:	179
Average score:	0 (0 votes)
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Secondary language:	English
Secondary title:	p-adic norms and p-adic numbers
Secondary abstract:	We can construct rational numbers Q as a quotient set of pairs (a, b) where a and b are integers or we can define fractions a/b, where b is not equal to 0. Two fractions a/b and c/d represent the same number if and only if ad=bc. In this thesis, we firstly describe absolute values on rational numbers Q: usual absolute value, trivial absolute value and p-adic absolute value. Then we prove the Ostrowski theorem, which says that every non-trivial absolute value on Q is equivalent to either usual absolute value or the p-adic absolute value for some prime number p. The metric space of rational numbers Q is complete with respect to the trivial absolute value. We know, however, that rational numbers Q are not complete with respect to the usual absolute value. We can extend the space of rational numbers Q with respect to the usual absolute value to get the space of real numbers, which is a complete metric space. If we take any p-adic absolute value on rational numbers Q instead of the usual absolute value, we get the metric space which is also not complete. The metric completion of this metric space is called p-adic numbers. We end the thesis with some characteristics of p-adic integers and prove the p-adic expansion of p-adic numbers.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. Ljubljana, Pedagoška fak., Dvopredmetni učitelj
Pages:	20 str.
ID:	8680355