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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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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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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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