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p-adic norms and p-adic numbers

Ida Femc (2014) p-adic norms and p-adic numbers. Diploma thesis.

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    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.

    Item Type: Thesis (Diploma thesis)
    Keywords: rational numbers, absolute value, p-adic absolute value, real numbers, p-adic numbers, p-adic integers
    Number of Pages: 20
    Language of Content: Slovenian
    Mentor / Comentors:
    Mentor / ComentorsIDFunction
    izr. prof. dr. Marko SlaparMentor
    Link to COBISS: http://www.cobiss.si/scripts/cobiss?command=search&base=50126&select=(ID=10203977)
    Institution: University of Ljubljana
    Department: Faculty of Education
    Item ID: 2414
    Date Deposited: 03 Oct 2014 13:57
    Last Modified: 03 Oct 2014 14:00
    URI: http://pefprints.pef.uni-lj.si/id/eprint/2414

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