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master's thesis
p-adski brojevi i algebarska proširenja
Zagreb: University of Zagreb, Faculty of Science, 2022. urn:nbn:hr:217:141590
Šakić, Tina
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Šakić, T. (2022). p-adski brojevi i algebarska proširenja (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:141590

Šakić, Tina. "p-adski brojevi i algebarska proširenja." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:141590

Šakić, Tina. "p-adski brojevi i algebarska proširenja." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:141590

Šakić, T. (2022). 'p-adski brojevi i algebarska proširenja', Master's thesis, University of Zagreb, Faculty of Science, accessed 04 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:141590

Šakić T. p-adski brojevi i algebarska proširenja [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2022 [cited 2024 November 04] Available at: https://urn.nsk.hr/urn:nbn:hr:217:141590

T. Šakić, "p-adski brojevi i algebarska proširenja", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:217:141590

Metadata
Titlep-adski brojevi i algebarska proširenja
Title (english)p-adic numbers and algebraic expansions
AuthorTina Šakić
MentorIgor Ciganović (mentor)
Committee memberIgor Ciganović (predsjednik povjerenstva)
Committee memberNikola Sandrić (član povjerenstva)
Committee memberSiniša Slijepčević (član povjerenstva)
Committee memberFranka Miriam Brückler (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2022-09-28, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom smo se radu bavili proučavanjem p-adskih brojeva koje je prvi puta opisao njemački matematičar Kurt Hensel 1897. godine. Skup p-adskih brojeva opisali smo u kontekstu teorije skupova, algebre i metričkoga prostora. Prvo smo krenuli od prstena \(\mathbb{Z}p\) p-adskih cijelih brojeva, a budući da smo pokazali da je \(\mathbb{Z}p\) integralna domena, proširili smo \(\mathbb{Z}p\) na polje razlomaka koje smo označavali sa \(\mathbb{Q}p\). Vidjeli smo da je na \(\mathbb{Q}p\) moguće definirati ultrametriku. Iskazali smo i dokazali jednu od osnovnih tvrdnji vezanu za p-adske brojeve, Henselovu lemu, koja govori da možemo doći do jedinstvenog rješenja jednadžbe P(x) = 0 za polinom P ∈ \(\mathbb{Z}p\) [x]. Vidjeli smo da je Henselova lema usko vezana na Newtonovu metodu pronalaženja aproksimacije nultočki. Dotaknuli smo se kvadratnih proširenja polja \(\mathbb{Q}p\) kao direktne posljedice Henselove leme. Na kraju smo proučavali algebarska proširenja p-adskih brojeva i vidjeli primjer konačnog Galoisovog proširenja.
Abstract (english)
In this master’s thesis we study p-adic numbers which were first described by Kurt Hensel, German mathematician, in 1897. We describe the set of p-adic numbers in context of set theory, algebra and metric spaces. First we observe the ring of p-adic integers, denoted by \(\mathbb{Z}p\). Ring \(\mathbb{Z}p\) is shown to be an integral domain, so it’s possible to define the field of fractions of \(\mathbb{Z}p\), which we denote by \(\mathbb{Q}p\). Also, we define an ultrametrics on \(\mathbb{Q}p\). We give a proof of Hensel’s lemma, one of the fundamental statements of p-adic theory, which shows that there is an unique solution of P(x) = 0 for x ∈ \(\mathbb{Z}p\) and P ∈ \(\mathbb{Z}p\) [x]. Also, we observe some connections between Newton’s method for finding an approximate solutions for f (x) = 0, and Hensel’s lemma. As direct consequence of Hensel’s lemma we observe quadratic extensions of \(\mathbb{Q}p\). In the very end of this paper, we study algebraic extensions of p-adic numbers and show an example of a finite Galois extension.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:141590
Study programmeTitle: Finance and Business Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2022-11-08 13:12:17