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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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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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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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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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