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Solving diophantine equations by factorization in number fields
Hrnčiar, Maroš ; Kala, Vítězslav (advisor) ; Příhoda, Pavel (referee)
Title: Solving diophantine equations by factorization in number fields Author: Bc. Maroš Hrnčiar Department: Department of Algebra Supervisor: Mgr. Vítězslav Kala, Ph.D., Mathematical Institute, University of Göttingen Abstract: The question of solvability of diophantine equations is one of the oldest mathematical problems in the history of mankind. While different approaches have been developed for solving certain types of equations, this thesis predo- minantly deals with the method of factorization over algebraic number fields. The idea behind this method is to express the equation in the form L = yn where L equals a product of typically linear factors with coefficients in a particular number field. Provided that several assumptions are met, it follows that each of the factors must be the n-th power of an element of the field. The structure of number fields plays a key role in the application of this method, hence a crucial part of the thesis presents an overview of algebraic number theory. In addition to the general theoretical part, the thesis contains all the necessary computations in specific quadratic and cubic number fields describing their basic characteristics. However, the main objective of this thesis is solving specific examples of equati- ons. For instance, in the case of equation x2 + y2 = z3 we...
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Universal quadratic forms over number fields
Svoboda, Josef ; Kala, Vítězslav (advisor) ; Hejda, Tomáš (referee)
The aim of this work is to study universal quadratic forms over biquadratic fields. In the thesis we define biquadratic fields and describe their structure. In particular, we study some distinguished (totally positive and aditively indecomposable) elements, their norms and traces. Then we describe the theory of universal quadratic forms and use special elements to find a lower bound for the number of variables of a universal quadratic form over some biquadratic fields.
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Composition of quadratic forms over number fields
Zemková, Kristýna ; Kala, Vítězslav (advisor) ; Francírek, Pavel (referee)
The thesis is concerned with the theory of binary quadratic forms with coefficients in the ring of algebraic integers of a number field. Under the assumption that the number field is of narrow class number one, there is developed a theory of composition of such quadratic forms. For a given discriminant, the composition is determined by a bijection between classes of quadratic forms and a so-called relative oriented class group (a group closely related to the class group). Furthermore, Bhargava cubes are generalized to cubes with entries from the ring of algebraic integers; by using the composition of quadratic forms, the composition of Bhargava cubes is proved in the generalized case. 1
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Universal quadratic forms over number fields
Svoboda, Josef ; Kala, Vítězslav (advisor) ; Hejda, Tomáš (referee)
The aim of this work is to study universal quadratic forms over biquadratic fields. In the thesis we define biquadratic fields and describe their structure. In particular, we study some distinguished (totally positive and aditively indecomposable) elements, their norms and traces. Then we describe the theory of universal quadratic forms and use special elements to find a lower bound for the number of variables of a universal quadratic form over some biquadratic fields.
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Solving diophantine equations by factorization in number fields
Hrnčiar, Maroš ; Kala, Vítězslav (advisor) ; Příhoda, Pavel (referee)
Title: Solving diophantine equations by factorization in number fields Author: Bc. Maroš Hrnčiar Department: Department of Algebra Supervisor: Mgr. Vítězslav Kala, Ph.D., Mathematical Institute, University of Göttingen Abstract: The question of solvability of diophantine equations is one of the oldest mathematical problems in the history of mankind. While different approaches have been developed for solving certain types of equations, this thesis predo- minantly deals with the method of factorization over algebraic number fields. The idea behind this method is to express the equation in the form L = yn where L equals a product of typically linear factors with coefficients in a particular number field. Provided that several assumptions are met, it follows that each of the factors must be the n-th power of an element of the field. The structure of number fields plays a key role in the application of this method, hence a crucial part of the thesis presents an overview of algebraic number theory. In addition to the general theoretical part, the thesis contains all the necessary computations in specific quadratic and cubic number fields describing their basic characteristics. However, the main objective of this thesis is solving specific examples of equati- ons. For instance, in the case of equation x2 + y2 = z3 we...
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Supporting algorithms of number field sieve
Skoková, Adéla ; Drápal, Aleš (advisor) ; Příhoda, Pavel (referee)
Title: Supporting algorithms of number field sieve Author: Adéla Skoková Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this work we study the first part of the algorithm of number field sieve, generating of polynomials. At first we describe all the algorithm of the sieve for understanding of the role of polynomials and their impact on the entire algorithm. Then we present their characteristics and evaluation. The last part is about the most effective know algorithms of generating polynomials, invented by Thorsen Klinjung. The second Kleinjung algoritm was also programmed. Keywords: GNFS, Number sieve, Number field, Kleinjung algorithm Powered by TCPDF (www.tcpdf.org)
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Supporting algorithms of number field sieve
Skoková, Adéla ; Drápal, Aleš (advisor) ; Příhoda, Pavel (referee)
Title: Supporting algorithms of number field sieve Author: Adéla Skoková Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this work we study the first part of the algorithm of number field sieve, generating of polynomials. At first we describe all the algorithm of the sieve for understanding of the role of polynomials and their impact on the entire algorithm. Then we present their characteristics and evaluation. The last part is about the most effective know algorithms of generating polynomials, invented by Thorsen Klinjung. Keywords: GNFS, Number sieve, Number field, Kleinjung algorithm
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