Gelfond-Schneiderjev izrek

magistrsko delo

Dejan Ozebek (Author), Marko Slapar (Mentor), Luka Boc Thaler (Co-mentor)

Abstract

V magistrskem delu dokažemo iracionalnosti števil e, π 2 in √n m, kjer m, n ∈ N in √n m ∈/ N, ki potekajo s protislovjem. Poleg tega dokažemo tudi transcendentnost števil e in π, ki imata nekoliko različen potek dokaza, saj se dokaz transcendentnosti števila π posluži tudi teorije elementarnih simetričnih polinomov. V zadnjem poglavju predstavimo še dokaz Gelfond Schneiderjevega izreka, o transcendentnosti števil oblike α β , kjer je α 6= 0, 1 in β iracionalno število. Gre namreč za enega redkih zapisov dokaza Gelfond Schneiderjevega izreka v slovenskem jeziku, saj se ta pojavi le še v diplomskem delu iz leta 1978 [6]. Ob koncu še naštejemo nekaj posledic tega izreka in hkrati navedemo nekaj avtorjev, ki so nadaljevali z raziskovanjem transcendentnih števil.

Keywords

algebraična števila;transcendentna števila;Gelfond-Schneiderjev izrek;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[D. Ozebek]
UDC:	512.6(043.2)
COBISS:	88148227
Views:	133
Downloads:	11
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Gelfond-Schneider theorem
Secondary abstract:	In this masters thesis we prove by contradiction the irrationality of the numbers e, π 2 , and √n m, where m, n ∈ N and √n m ∈/ N. Alongside we also prove the transcendence of the numbers e and π, which have distinctly different proofs, as the proof for the transcendence of the number π makes use of the theory of elementary symmetric polynomials. In the last chapter, we also prove the main theorem of this work, which is the Gelfond-Schneider theorem on the transcendence of numbers of the form α β , where α 6= 0, 1 and β is irrational. This work is one of the rare translations of this theorem in Slovene; the only other translation appears in an undergraduate thesis from 1978 [6]. We conclude the thesis by listing several consequences of the theorem and a number of authors who advanced the work in transcendental number theory.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje, Predmetno poučevanje
Pages:	45 str.
ID:	13889772