
Spectrum problem
Ježil, Ondřej ; Krajíček, Jan (advisor) ; Šaroch, Jan (referee)
We study spectra of firstorder sentences. After providing some interesting examples of spectra we show that the class of spectra is closed under some simple settheoretic and algebraic operations. We then define a new class of definable operations generalizing the earlier constructions. Our main result is that the class of these operations is, in a suitable technical sense, closed under a form of iteration. This in conjunction with Cobham's characterisation of FP offers a new proof of Fagin's theorem and also of the JonesSelman characterisation of spectra as NE sets. 1

 

Testing the projectivity of modules
Matoušek, Cyril ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the problem of the existence of test modules for the projectivity. A right Rmodule is said to be a test module if it holds for every right Rmodule M that M is projective whenever T ∈ M⊥ . We show that test modules exist over right perfect rings, although their existence is not provable in ZFC in case of nonright perfect rings. In order to prove this, we use Shelah's uni formization principle, which is independent of the axioms of ZFC. Furthermore, we show that test modules exist over rings of finite global dimension assuming the weak diamond principle, which is also independent of ZFC. 1


Primes in discretely ordered quasiEuclidean domains
Sgallová, Ester ; Šaroch, Jan (advisor) ; Glivická, Jana (referee)
This thesis studies discretely ordered quasiEuclidean domains. The goal is to study primes and prime pairs in them and to answer the question, whether there can be a cofinal set of them. The first construction gives a domain that does not have a cofinal set primes. Another construction builds a principal ideal domain, which has a cofinal set of primes, but no two distinct nonstandard primes differ by a natural number, so there is not a cofinal set of prime pairs. Furthermore, the thesis describes a construction of a principal ideal domain, whitch has a cofinal set of prime apairs for any even positive integer a. 1


Max rings
Beneš, Daniel ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Topic of this thesis is max rings, which are the rings, whose nonzero modu les have maximal submodules. At the begining we prove a characterization of commutative max rings as rings with Tnilpotent Jacobson radical and von Ne umann regular factor ring of the Jacobson radical. Our next concern are group rings, where we describe all commutative group rings, that are max. These are the group rings, that are composed from a commutative max ring and an abelian torsion group, where is finitely many elements of order pn for p not invertible in the ring. Finally we use this characterization to construct noncommutative group rings, which are max but not perfect.


Multilinear Maps Over the Integers
Havránek, František ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
The thesis aims to describe the [CLT15] scheme, which is based on the Diffie Hellman scheme and uses multilinear maps over integers. This scheme enables an exchange of a key among several participants. The level κ scheme (using a κlinear map) enables the exchange of a key among κ + 1 participants. The thesis introduces the basic terms, describes the needed theory, the base of which is the Chinese Remainder Theorem, and also the preparation and usage of the scheme. The correctness of the scheme is proved as well and the related requirements on the basic parameters are discussed.


Invariant theory for finite groups
Žurav, Martin ; Šťovíček, Jan (advisor) ; Šaroch, Jan (referee)
The prime goal of this thesis is to give a decent introduction to the theory of invariants for finite groups. We begin our characterisation with symmetric polynomials and their fundamental properties. In particular, we study the ring of symmetric polynomials and we prove that it is finitely generated by elementary symmetric functions. Then we deal with some of the criteria for a polynomial to be symmetric. In the second part, we generalise these ideas for any finite subgroup of GL(n,k). We define an action of a finite linear group on k[x_1,...,x_n] and consider polynomials that are invariant under such action. We show that they form a ring which is always finitely generated, as follows from the Noether's bound theorem. At the end, we describe the ring of invariants and relations among its generators more profoundly.


Fast multiplication in the field GF(2n)
Bajtoš, Marek ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Title: Fast multiplication in the field GF(2n ) Author: Marek Bajtoš Department: Department of Algebra Supervisor: doc. Mgr. et Mgr. Žemlička Jan, Ph.D., Department of Algebra Abstract: In this bachelor thesis we research how to optimize multiplication with a fixed element of finite field which can be useful for implementation of crypto graphic algorithms in lightweight cryptography. We will represent effectivity of multiplication by number of XOR operation needed for implementation of matrix which represent some fixed element of finite field. We prove that some matrix re presents multiplication with some element of finite field if and only if the minimal polynomial of matrix is irreducible. We also prove theorems describing conditi ons which matrix must satisfy so matrix can be implemented with only 1 or 2 XOR operations. At the end of the thesis we show construction of circulant MDS matrices which uses elements of finite field with low XOR count so they can be easily implemented. Keywords: lightweight cryptography, finite field, XOR, MDS matrix


Tilting theory of commutative rings
Hrbek, Michal ; Trlifaj, Jan (advisor) ; Herbera Espinal, Dolors (referee) ; Šaroch, Jan (referee)
The thesis compiles my contributions to the tilting theory, mainly in the set ting of a module category over a commutative ring. We give a classification of tilting classes over an arbitrary commutative ring in terms of data of geometrical flavor  certain filtrations of the Zariski spectrum. This extends and connects the results known previously for the noetherian case, and for Prüfer domains. Also, we show how the classes can be expressed using the local and Čech homology the ory. For 1tilting classes, we explicitly construct the associated tilting modules, generalizing constructions of Fuchs and Salce. Furthermore, over any commuta tive ring we classify the silting classes and modules. Amongst other results, we exhibit new examples of cotilting classes, which are not dual to any tilting classes  a phenomenon specific to nonnoetherian rings. 1


Decidability of the theory of commutative groups
Čech, František ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis will be demonstrated proof of decidability of theory of commu tative groups. This result was already shown in year 1955 by author W.Szmielew. However proof shown here takes different path. Result will by shown with use of results from theory of modules and theory of modeles prooved in article by M. Ziegler Model theory of modules. Final part of proof follows proof shown in article The elementary theory of Abelian groups by P. C. Eklofa and E. R. Fishera. 1
