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The use of Computers in the Number Theory
Konečný, Zdeněk ; Karásek, Jiří (referee) ; Skula, Ladislav (advisor)
PARI/GP is a relatively obscure mathematical software that was designed especially for quick calculations in the number theory, but which has also found its application in other areas of mathematics. This work gives an overview of the basic commands of PARI/GP, and some simple examples to show their potential use. PARI/GP is then used for looking for large primes of special forms.
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General Codes with m Marks
Holešovský, Jan ; Hrdina, Jaroslav (referee) ; Skula, Ladislav (advisor)
This bachelor's thesis is concerned with results of error-correcting codes theory, which deals with detection and correction of errors, that arise during communication by means of these codes. The aim of this thesis is the explanation of the theory above in absolute generality, followed by detail view of some significant codes. Using linear algebra over finite fields, we will introduce an error-corecting code like a set with structure, whose characters considerably simplify the detection and correction of errors. The knowledge, that was acquired for general codes, is applied to well-known binary codes at the end of the thesis (ie. Hamming codes and Golay code). With these codes are demonstrated their properties, that sort these codes to the most important binary codes.
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Fundamentals of quadratic fields
Ivičič, Vojtěch ; Karásek, Jiří (referee) ; Skula, Ladislav (advisor)
The aim of the thesis is to describe the fundamentals of quadratic fields. In the first part integer and polynomial arithmetic is mentioned. The main part discusses the topic of quadratic field, quadratic integer, Gaussian field and Gaussian integer. The final part deals with quadratic fields without unique factorization and a solution of the problem through ideals.
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Discretely normed orders of quaternionic algebras
Horníček, Jan ; Skula, Ladislav (referee) ; Kureš, Miroslav (advisor)
Tato práce shrnuje autorův výzkum v oblasti teorie kvaternionových algeber, jejich izomorfismů a maximálních řádů. Nový úhel pohledu na tuto problematiku je umožněn využitím pojmu diskrétní normy. Za hlavní výsledky práce je možná považovat důkaz jednoznačnosti diskrétní normy pro celá čísla, kvadratická rozšíření těles a řády kvaternionových algeber. Dále větu, která umožňuje mezi dvěma kvaternionovými algebrami konstruovat izomorfismy explicitně vyjádřené v maticovém tvaru. A v neposlední řadě důkaz existence nekonečně mnoha různých maximálních řádů kvaternionové algebry. Výsledky uvedené v této diplomové práci budou dále publikovány ve vědeckém článku.
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Discretely normed orders of quaternionic algebras
Horníček, Jan ; Skula, Ladislav (referee) ; Kureš, Miroslav (advisor)
Tato práce shrnuje autorův výzkum v oblasti teorie kvaternionových algeber, jejich izomorfismů a maximálních řádů. Nový úhel pohledu na tuto problematiku je umožněn využitím pojmu diskrétní normy. Za hlavní výsledky práce je možná považovat důkaz jednoznačnosti diskrétní normy pro celá čísla, kvadratická rozšíření těles a řády kvaternionových algeber. Dále větu, která umožňuje mezi dvěma kvaternionovými algebrami konstruovat izomorfismy explicitně vyjádřené v maticovém tvaru. A v neposlední řadě důkaz existence nekonečně mnoha různých maximálních řádů kvaternionové algebry. Výsledky uvedené v této diplomové práci budou dále publikovány ve vědeckém článku.
|
|
|
Fundamentals of quadratic fields
Ivičič, Vojtěch ; Karásek, Jiří (referee) ; Skula, Ladislav (advisor)
The aim of the thesis is to describe the fundamentals of quadratic fields. In the first part integer and polynomial arithmetic is mentioned. The main part discusses the topic of quadratic field, quadratic integer, Gaussian field and Gaussian integer. The final part deals with quadratic fields without unique factorization and a solution of the problem through ideals.
|
|
| |
|
|
General Codes with m Marks
Holešovský, Jan ; Hrdina, Jaroslav (referee) ; Skula, Ladislav (advisor)
This bachelor's thesis is concerned with results of error-correcting codes theory, which deals with detection and correction of errors, that arise during communication by means of these codes. The aim of this thesis is the explanation of the theory above in absolute generality, followed by detail view of some significant codes. Using linear algebra over finite fields, we will introduce an error-corecting code like a set with structure, whose characters considerably simplify the detection and correction of errors. The knowledge, that was acquired for general codes, is applied to well-known binary codes at the end of the thesis (ie. Hamming codes and Golay code). With these codes are demonstrated their properties, that sort these codes to the most important binary codes.
|
|
|
The use of Computers in the Number Theory
Konečný, Zdeněk ; Karásek, Jiří (referee) ; Skula, Ladislav (advisor)
PARI/GP is a relatively obscure mathematical software that was designed especially for quick calculations in the number theory, but which has also found its application in other areas of mathematics. This work gives an overview of the basic commands of PARI/GP, and some simple examples to show their potential use. PARI/GP is then used for looking for large primes of special forms.
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