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Semilattices and indecomposable elements
Kuděj, Martin ; Kala, Vítězslav (advisor) ; Korbelář, Miroslav (referee)
This thesis concerns the theory of semilattices, which are non-trivial discrete additive submonoids of Rn , which are contained in a cone. Special emphasis is on their indecomposable elements. The most important example of semilattices is derived from real quadratic number fields, which involves the most parts of the thesis and all indecomposable elements of such semilattices are characterised in two ways. That includes using various tools from number theory, mainly con- tinued fractions, their corresponding semiconvergents and their approximation properties, Farey pairs, but also some tools from algebraic number theory. The final part of the thesis concerns the upper bound of the norm of indecompos- able elements in a semilattice, derived from the Minkowski embedding of the corresponding number field. 1
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Universal quadratic forms over orders in number fields
Krásenský, Jakub ; Kala, Vítězslav (advisor) ; Nebe, Gabriele (referee) ; Becher, Karim Johannes (referee)
This thesis studies quadratic forms and lattices over rings of integers in number fields, and, to some extent, over non-maximal orders as well. The main focus is on universality of forms and lattices, and on the connected notion of the Pythagoras number. We mostly study totally positive definite forms and lattices over totally real fields, as this is arguably the most difficult case with very unpredictable behaviour. In some chapters we develop a general theory valid for fields of all degrees - in particular when we study the quadratic Waring's problem -, while in others we present important detailed results for families of fields in low degrees. By this we fulfill several purposes: We obtain interesting results, directly in line with questions studied by the likes of Siegel; we illustrate and further develop a multitude of techniques which could be applied in other situations; and we formulate conjectures and questions which can stimulate further research. Particular focus is put on real biquadratic fields; the first two chapters significantly advance the theory of integral quadratic forms over them. 1
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RSA in number fields and on lattices
Kucka, Filip Miroslav ; Kala, Vítězslav (advisor) ; Šůstek Vyhnalová, Sára (referee)
This thesis is focused on the RSA algorithm in number fields and on lattices. Specif- ically, we extend the work the authors Zheng and Liu in their article High Dimensional RSA. In the thesis we precisely describe all the theory required theory with theorems and examples using mostly Algebraic number theory and lattice theory. In the second chapter, we create the RSA only in number fields, we discuss its problems and the ne- cessity of lattices. In the third chapter, we precisely describe and prove properties of ideal matrices, we define the vector multiplication in Rn and at the end ve prove the ring isomorphism K ≃ Qn ≃ M∗ Q. In the fourth chapter, we prove the ring isomorphism Z[x]/(mθ(x)) ≃ OK ≃ Zn ≃ M∗ Z, we define ideal lattices and we create all the required theory over lattices for RSA. The last chapter consists of the complete RSA algorithm in number fields and on lattices and example. 1
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Cyclotomic extensions and the Kronecker-Weber theorem
Jarrahová, Veronika ; Kala, Vítězslav (advisor) ; Francírek, Pavel (referee)
In the thesis, we prove the Kronecker-Weber theorem, which states that every abelian extension of the field of rational numbers is a subfield of some cyclotomic field. This theorem is traditionally proved using class field theory, but we will use an alternative relatively elementary proof using Galois theory and algebraic number theory. We will first introduce the necessary theory and show the new definitions with an example. The key part of the whole proof will be to prove the Kronecker-Weber theorem for abelian expansions of prime power degree, where only this prime ramifies. Then, we can prove relatively easily that the theorem holds for general abelian extensions. 1
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The structure of generalized Pythagorean triples
Hlavinková, Simona ; Kala, Vítězslav (advisor) ; Krásenský, Jakub (referee)
The motivation for our thesis is to describe generalized of Phytagorean triples. We convert this problem into the problem of finding a solution of the equation |x2 +Dy2 | = z2 . The goal of this thesis is to prove in detail the structure and the number of solutions of the equation |x2 + Dy2 | = z2 for −D ≡ 2, 3 (mod 4) and square-free. The proofs of lemmas are proved by using properties of ideal class group of number field Q[ √ −D]. We first prove a lemma that gives us the necessary conditions for the existence of a solution. We describe the connection between uniqueness, respectively ambiguity of the solution and the choice of D. The most important step of the proof is to express the solution in a special form. We also give examples of structure of ideal class group of various number fields. 1
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Partitions of totally positive elements in real quadratic fields
Stern, David ; Kala, Vítězslav (advisor) ; Gil Muñoz, Daniel (referee)
We consider the additive semigroup O+ K(+) of totally positive integers in a real quadratic field K = Q( √ D). We define on O+ K(+) the partition function pK(α) and de- velop an algorithm for computing pK(α) for different square-free D and different α ∈ O+ K. We then investigate the behaviour of pK(α), characterizing the square-free numbers D for which pK(α) attains the numbers 1 through 5. Finally, we prove a sufficient condition for the number 6 to be attainable by pK(α). 1
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Adeles and class fields
Tížková, Bára ; Kala, Vítězslav (advisor) ; Gajović, Stevan (referee)
The first aim of the thesis is to study the ring of adèles and the group of idèles and work out their topology in detail. We explain the relation between the restricted product topology and other topologies which might seem natural on these objects. Further, we study their compactness properties. The second aim of the thesis is to summarize the main results of class field theory, both in the language of ideals and of idèles, and to provide examples illustrating new notions and concepts. Roughly speaking, class field theory describes all abelian extensions of a number field in terms of some "inner arithmetic" of the field. First, we demonstrate how the description works on two particular types of abelian extensions and then generalize the notions for any abelian extension. In order to present the general class field theory in a clear and straightforward manner, we unify the content of various sources and literature. The classical approach via ideals is more natural; however, some inconveniences arise when we have to take into account which primes ramify in the extension. This is handled by the idèlic approach described in the last chapter. 1
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Fourier transform on polytopes and tiling with rectangles
Couf, František ; Kala, Vítězslav (advisor) ; Čech, Martin (referee)
The aim of this thesis is a detailed exposition of the proof of a theorem on nice intervals in d dimensions and a theorem on the properties of harmonic intervals. We introduce first the notions of a nice rectangle and a tiling of a set. Then we extend the notion of a nice rectangle to that of a nice d-dimensional interval. We subsequently prove the main theorem (a closed interval tiled by nice closed intervals is also nice) in d dimensions. Then we define harmonic intervals and prove in detail several important theorems on tiling by harmonic intervals. We illustrate their assumptions with examples which demonstrate their importance. We show the connection between the notions of a harmonic interval and a multiple of an interval for intervals with integral edges at the end of the last chapter. 1
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Geometric solution of quadratic diophantine equations
Lněničková, Daniela ; Kala, Vítězslav (advisor) ; Krásenský, Jakub (referee)
The main goal of the work is to summarize and generalize a method for solving quadratic Diophantine equations. We transform the problem of finding the solution of Diophantine equations to finding the intersections of lines and a given quadric. The theory works over a general field and is able to solve equations leading to quadrics of n variables. We then apply the theory to solve some examples, namely the search for Pythagorean triplets over Gaussian integer and the equation leading to the hyperboloid, where we use our generalization. 1
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Subfields of number field extensions and quadratic forms
Doležálek, Matěj ; Kala, Vítězslav (advisor) ; Gil Muñoz, Daniel (referee)
A number of recent results give constructions of totally real number fields of specific degrees that do not admit universal quadratic forms of small rank. Given a totally real number field L that is known to have a certain lower bound on the rank of universal quadratic forms, one may try to construct extensions of L that also satisfy this bound. In this thesis, we present a way of constructing such an extension as the compositum of L and some suitable number field K. The construction relies on inequalities involving traces and discriminants in number fields and controlling the subfields of KL using Galois correspondence, which then leads to examining subgroups in direct products of groups. 1
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