Hassejevo načelo

magistrsko delo

Saša Trstenjak (Author), Urban Jezernik (Mentor)

Abstract

V delu s pomočjo Hassejevega načela obravnavamo obstoj racionalnih ničel homogenih kvadratnih polinomov z racionalnimi koeficienti. Preko inverzne limite definiramo $p$-adična števila ${\mathbb Q}_p$ in obravnavamo rešljivost enačb v množici ${\mathbb Q}_p$. Nato definiramo Legendrov simbol in Hilbertov simbol, obravnavamo kvadrate v p-adičnih številih ter dokažemo Hassejevo načelo za homogene kvadratne polinome največ treh spremenljivk. V nadaljevanju si podrobneje pogledamo splošne kvadratne forme in dokažemo Hassejevo načelo še za homogene kvadratne polinome štirih in več spremenljivk. Navedemo še nekaj primerov polinomskih enačb višjih stopenj, ki ne ustrezajo Hassejevemu načelu. Na koncu si na primeru kubičnih form treh spremenljivk pogledamo, kolikšen delež jih ustreza Hassejevemu načelu in kolikšen delež ga ovrže.

Keywords

matematika;p-adična števila;kvadratna forma;Hassejevo načelo;Hilbertov simbol;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[S. Trstenjak]
UDC:	511
COBISS:	119343619
Views:	852
Downloads:	92
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Hasse principle
Secondary abstract:	In this thesis we explore the Hasse principle and use it to consider the existence of rational zeros of homogeneous quadratic polynomials with rational coefficients. We define $p$-adic numbers ${\mathbb Q}_p$ as an inverse limit and consider solvability of equations in the set ${\mathbb Q}_p$. We then define the Legendre symbol and the Hilbert symbol, consider p- adic squares, and prove the Hasse principle for homogeneous quadratic polynomials of up to three variables. Then we take a closer look at general quadratic forms and prove the Hasse principle for homogeneous quadratic polynomials of four or more variables. Next, we give a few examples of higher-degree polynomial equations that do not satisfy the Hasse principle. Finally, in the case of cubic forms of three variables, we look at what proportion of them satisfy the Hasse principle and what proportion do not.
Secondary keywords:	mathematics;p-adic numbers;quadratic form;Hasse principle;Hilbert symbol;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika
Pages:	IX, 78 str.
ID:	16306604