Jednadžbe u p-adskim brojevima

master's thesis

• Author: Dragana Dublec
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Željka Milin-Šipuš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-11-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome diplomskom radu bavimo se \(p\)-adskim brojevima, pri čemu je naglasak na jednadžbama s \(p\)-adskim brojevima. U prvom poglavlju uvodimo \(p\)-adske cijele i \(p\)-adske brojeve promatrajući analogiju između prstena cijelih brojeva, \(\mathbb{Z}\), zajedno s poljem razlomaka \(\mathbb{Q}\) i prstenom polinoma s koeficijentima iz skupa kompleksih brojeva, \(\mathbb{C}[X]\), zajedno s poljem razlomaka od \(\mathbb{C}[X]\). Zatim u idućem potpoglavlju se bavimo \(p\)-adskom apsolutnom vrijednošću u kojem dokazujemo i teorem Ostrowskog koji povezuje svaku netrivijalnu apsolutnu vrijednost na polju \(\mathbb{Q}\) i \(p\)-adsku apsolutnu vrijednost. Na kraju prvog poglavlja dajemo kratki pregled "dobrih" i "loših" svojstava \(p\)-adskih brojeva uspoređujući ih s racionalnim brojevima. U drugom poglavlju se prvo bavimo računanjem u \(\mathbb{Q}_p\). Pri čemu posebnu pažnju dajemo postojanju drugog korijena u polju \(\mathbb{Q}_p\). Već nam to predstavlja lagani uvod za jednadžbe s \(p\)-adskim brojevima. Nakon toga rješavamo kongruencije modulo \(p^n\) koje sad već dovoljno "bliske" jednadžbama u \(p\)-adskim brojevima, da bi ih mogli smatrati jednadžbama s \(p\)-adskim brojevima. Zatim dajemo neke važne rezultate o rješenjima jednadžbi s \(p\)-adskim brojevima čiji su koeficijenti \(p\)-adski cijeli brojevi. Što je bio i cilj ovoga rada. U zadnjem poglavlju dajemo kratki sažetak u kojem je vidljiva primjena \(p\)-adskih brojeva.

Abstract (english)

In this thesis we deal with the \(p\)-adic numbers, with the emphasis on equations with \(p\)-adic numbers. In the first chapter, we introduce the \(p\)-adic integers and \(p\)-adic numbers by looking at the analogy between the ring of integers, \(\mathbb{Z}\), together with the field of fractions \(\mathbb{Q}\) and the ring of polynomials with coefficients from a set of complex numbers, \(\mathbb{C}[X]\), together with the field of fractions of \(\mathbb{C}[X]\). Then in the next subsection we deal with the \(p\)-adic absolute value in which we prove a theorem and Ostrowski that connects each nontrivial absolute value of the field \(\mathbb{Q}\) and the \(p\)-adic absolute value. At the end of the first chapter we give a brief outline of "good" and "bad" properties of \(p\)-adic numbers by comparing them with rational numbers. In the second chapter, we first deal with calculations in \(\mathbb{Q}_p\). Special attention is given to the existence of square roots in the field \(\mathbb{Q}_p\). For us, this represent a slight introduction to the equation with \(p\)-adic numbers. After that we solve congruences modulo \(p^n\) whichare now sufficiently "close" to equations in \(p\)-adic numbers to considered them equations in \(p\)-adic numbers. Then we give some important results about solutions of equations in \(p\)-adic numbers with coefficients being \(p\)-adic integers. This was the aim of this work. The last chapter gives a brief summary in which is visible on application of \(p\)-adic numbers.

• Keywords: \(p\)-adski brojevi; jednadžbe s \(p\)-adskim brojevima; Henselova analogija; primjena \(p\)-adskih brojeva
• Keywords (english): \(p\)-adic numbers; equations with \(p\)-adic numbers; Hensel’s analogy; application of \(p\)-adic numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:897113
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-03-21 13:15:40