Polinomi s celoštevilskimi vrednostmi

delo diplomskega seminarja

Lucija Udir (Author), David Dolžan (Mentor)

Abstract

V diplomskem delu je predstavljen kolobar Int$(\mathbb{Z})$, ki ga sestavljajo polinomi z racionalnimi koeficienti, ki za cela števila zavzemajo celoštevilske vrednosti. Ta kolobar ima drugačne lastnosti kot večina kolobarjev, ki jih preučujemo v komutativni algebri. Največ pozornosti smo posvetili dejstvu, da ima kolobar polinomov s celoštevilskimi vrednostmi lastnost dveh generatorjev. Znani dokazi te lastnosti so precej zapleteni, saj uporabljajo močne topološke argumente. V tem delu je predstavljen konstruktivni dokaz, ki uporablja osnovna algebraična orodja. Za lažje razumevanje smo definirali pojme, kot so kolobar, ideal, noetherski kolobar in Prüferjeva domena. Za pomoč pri dokazu lastnosti dveh generatorjev smo uporabili razširjen Evklidov algoritem, Skolemovo lastnost karakterizacije idealov z njihovimi ideali vrednosti ter druge potrebne trditve in leme. Skozi celotno diplomsko delo kolobar polinomov s celoštevilskimi vrednostmi primerjamo s kolobarjem polinomov s celoštevilskimi koeficienti in opisane lastnosti ponazorimo z zgledi.

Keywords

matematika;kolobarji;ideali;polinomi s celoštevilskimi vrednostmi;noetherski kolobarji;Prüferjeva domena;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[L. Udir]
UDC:	512
COBISS:	122673155
Views:	369
Downloads:	40
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary title:	Integer-Valued Polynomials
Secondary abstract:	In this thesis we introduce the ring Int$(\mathbb{Z})$, which consists of polynomials with rational coefficients that take integer values for integers. This ring has different properties from most of the rings studied in commutative algebra. We have focused on the fact that the polynomial with integer values has the property of two generators. The known proofs of this property are rather complicated, since they use strong topological arguments. In this paper we present a constructive proof that uses basic algebraic tools. For a better understanding, we define notions such as the ring, the ideal, Noethererian ring and the Pr ̈ufer domain. For the proof of the two-generator property, we have used the extended Euclidean algorithm, the Skolem property of characterising ideals by their ideals of values, and other necessary assertions and lemmas. Throughout the thesis, we compare the polynomial ring with integer values with the polynomial ring with integer coefficients and illustrate the described properties with examples.
Secondary keywords:	mathematics;rings;ideals;integer-valued polynomials;Noetherian rings;Prüfer domain;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	[25] str.
ID:	16542892