The p-adic numbers and Hensel's lemma
(2023) In Bachelor's Theses in Mathematical Sciences MATK11 20221Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- The thesis consists of four chapters, each focused on developing the knowledge and drawing new connections with the p-adic numbers and the notions that are well known from the usual analysis on the real topology.
In the first chapter, we introduce the basic structure of the $p$-adics, in connection to the conergent series and field axioms. Following that, in chapter two we focus on the defining the distance in the p-adic world, recalling the definition of the absolute value, valuation and exploring the metric space defined by them.
After introducing the distance, we will introduce the p-adic topology connected to it, recall some of the notions from the topology and explore more the p-adic field structure. Our considerations will be... (More) - The thesis consists of four chapters, each focused on developing the knowledge and drawing new connections with the p-adic numbers and the notions that are well known from the usual analysis on the real topology.
In the first chapter, we introduce the basic structure of the $p$-adics, in connection to the conergent series and field axioms. Following that, in chapter two we focus on the defining the distance in the p-adic world, recalling the definition of the absolute value, valuation and exploring the metric space defined by them.
After introducing the distance, we will introduce the p-adic topology connected to it, recall some of the notions from the topology and explore more the p-adic field structure. Our considerations will be rewarded by introducing the Hensel's lemma, centered on polynomials and solving congruence equations with their help. (Less) - Popular Abstract
- As the stereotype states, mathematicians tend to think of yet nonexistent problems and seek solution to them as a challenge. Such a rich and still expanding world of the p-adics, the main focus of this thesis, have arisen in this exact manner - with a question: what if one change 10 in decimal representation of a number into a prime? This simple idea led to the discovery of completely new mathematics, that, although follows the same schemes, produces some surprising twists.
Here, the reader shall gain knowledge about the basic laws of mathematics redefined in the p-adic setting. We will begin by introducing the necessary different branches of mathematics, starting from basic arithmetic properties, through topology and absolute values,... (More) - As the stereotype states, mathematicians tend to think of yet nonexistent problems and seek solution to them as a challenge. Such a rich and still expanding world of the p-adics, the main focus of this thesis, have arisen in this exact manner - with a question: what if one change 10 in decimal representation of a number into a prime? This simple idea led to the discovery of completely new mathematics, that, although follows the same schemes, produces some surprising twists.
Here, the reader shall gain knowledge about the basic laws of mathematics redefined in the p-adic setting. We will begin by introducing the necessary different branches of mathematics, starting from basic arithmetic properties, through topology and absolute values, finishing on the power series.
Therefore if the reader is looking for the answers to how is it possible that there are more $p$-adic numbers than the ones with base of 10, or why do only isosceles triangles exist in the p-adic space, this reading should not come as a disappointment. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9134124
- author
- Dabrowska, Justyna LU
- supervisor
- organization
- course
- MATK11 20221
- year
- 2023
- type
- M2 - Bachelor Degree
- subject
- keywords
- p-adic, p-adic numbers, number theory, hensel's lemma
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFMA-4150-2023
- ISSN
- 1654-6229
- other publication id
- 2023:K24
- language
- English
- id
- 9134124
- date added to LUP
- 2023-10-09 16:47:01
- date last changed
- 2023-10-09 16:47:01
@misc{9134124,
abstract = {{The thesis consists of four chapters, each focused on developing the knowledge and drawing new connections with the p-adic numbers and the notions that are well known from the usual analysis on the real topology.
In the first chapter, we introduce the basic structure of the $p$-adics, in connection to the conergent series and field axioms. Following that, in chapter two we focus on the defining the distance in the p-adic world, recalling the definition of the absolute value, valuation and exploring the metric space defined by them.
After introducing the distance, we will introduce the p-adic topology connected to it, recall some of the notions from the topology and explore more the p-adic field structure. Our considerations will be rewarded by introducing the Hensel's lemma, centered on polynomials and solving congruence equations with their help.}},
author = {{Dabrowska, Justyna}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor's Theses in Mathematical Sciences}},
title = {{The p-adic numbers and Hensel's lemma}},
year = {{2023}},
}