A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on finite Abelian groups show that Fuglede conjecture is false in high dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the finite groups ZN. Based on the Coven- Meyerowitz conjecture, it has been proved that if any spectral set in Z satisfies the the Coven-Meyerowitz properties, then every spectral set in R is a tile. We will present some of the main results related to integer tiles and give a self-contained description of the theory with detailed proofs.

We generalise the study of cyclotomic matrices - those with all eigenvalues in the interval [-2; 2] - from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined. We introduce the notion of `4-cyclotomic' matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. Six rings OQ( p d) for d = -1;-2;-3;-7;-11;-15 give rise to examples not found in the rational-integer case; in four (d = -1;-2;-3;-7) we recover infinite families as well as sporadic cases. For d = -15;-11;-7;-2, we demonstrate that a maximal cyclotomic graph is necessarily 4- cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.

We give explicit factorizations of $a$-cyclotomic polynomials of order $2^m 3$, $Q_{2^m3,a}(x)$, over a finite field $F_q$ with $q$ elements where $q$ is a prime power, $m$ is a nonnegative integer and $a$ is a nonnegative element of $F_q$. We use the relation between usual cyclotomic polynomials and $a$-cyclotomic polynomials. Factorizations split into eight categories according to $q \equiv \pm1$ (mod 4), $a$ and $-3$ are square in $F_q$. We find that the coefficients of irreducible factors are primitive roots of unity and in some cases that are related with Dickson polynomials.

This thesis establishes a framework for cellularity of algebras related to the Jones basic construction. The framework allows a uniform proof of cellularity of Brauer algebras, BMW algebras, walled Brauer algebras, partition algebras, and others. In this setting, the cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for this class of algebras, the cellular structures are compatible with restriction and induction of modules.

Orientador: Antonio Aparecido de Andrade / Banca: Clotilzio Moreira dos Santos / Banca: Edson Donizete de Carvalho / Resumo: O objetivo deste trabalho e demonstrar o Teorema de Kronecker-Weber de uma forma mais elementar, usando o artigo "An Elementary Proof of the Kronecker-Weber Theorem". O trabalho traz como aplicação uma fórmula para o cálculo do condutor de um corpo de números abeliano / Abstract: The objective of this work is to demonstrate the Kronecker-Weber Theorem in a form more elementary, using article "An Elementary Proof of the Kronecker-Weber Theorem". Application as the work gives a formula for calculating the conductor of a eld of numbers abelian. / Mestre

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomicprimality test, and shortly after, in 1981, a simplified and more efficient versionwas presented by H.W. Lenstra for the Bourbaki Seminar. Later, in 2008, ReneSchoof presented an updated version of Lenstra's primality test. This thesis presents adetailed description of the cyclotomic primality test as described by Schoof, along withsuggestions for implementation. The cornerstone of the test is a prime congruencerelation similar to Fermat's little theorem" that involves Gauss or Jacobi sumscalculated over cyclotomic fields. The algorithm runs in very nearly polynomial time.This primality test is currently one of the most computationally efficient tests and isused by default for primality proving by the open source mathematics systems Sageand PARI/GP. It can quickly test numbers with thousands of decimal digits.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-02-24Bitstream added on 2014-06-13T19:06:35Z : No. of bitstreams: 1 mendonca_acm_me_sjrp.pdf: 653758 bytes, checksum: 032e92ad4ceaecd8b97859ec301bd326 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho e demonstrar o Teorema de Kronecker-Weber de uma forma mais elementar, usando o artigo An Elementary Proof of the Kronecker-Weber Theorem. O trabalho traz como aplicação uma fórmula para o cálculo do condutor de um corpo de números abeliano / The objective of this work is to demonstrate the Kronecker-Weber Theorem in a form more elementary, using article An Elementary Proof of the Kronecker-Weber Theorem. Application as the work gives a formula for calculating the conductor of a eld of numbers abelian.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a finite field Fr have many properties in common. It is due to these properties that almost all the famous (multiplicative) number theoretic results over Z have analogues over A. In this thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules. We do this to survey and compare the analogues of cyclotomic polynomials, the size of their coefficients and cyclotomic extensions over the rational function field k = Fr(T). / AFRIKAANSE OPSOMMING: Dit is bekend dat Z, die ring van heelgetalle en A = Fr[T], die ring van polinome oor ’n eindige liggaam baie eienskappe in gemeen het. Dit is as gevolg van hierdie eienskappe dat feitlik al die bekende multiplikative resultate wat vir Z geld, analoë in A het. In hierdie tesis, fokus ons op die gebruik van hierdie analogie saam met die teorie van die Carlitz module. Ons doen dit om ’n oorsig oor die analoë van die siklotomiese polinome, hul koëffisiënte, en siklotomiese uitbreidings oor die rasionele funksie veld k = Fr(T).

Doctor of Philosophy / This thesis presents a study of the cyclotomic BMW algebras, introduced by Haring-Oldenburg as a generalization of the BMW (Birman-Murakami-Wenzl) algebras related to the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. The motivation behind the definition of the BMW algebras may be traced back to an important problem in knot theory; namely, that of classifying knots (and links) up to isotopy. The algebraic definition of the BMW algebras uses generators and relations originally inspired by the Kauffman link invariant. They are intimately connected with the Artin braid group of type A, Iwahori-Hecke algebras of type A, and with many diagram algebras, such as the Brauer and Temperley-Lieb algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics, and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, Haring-Oldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the Ariki-Koike algebra h_{n,k} is a quotient of B_n^k, in the same way the Iwahori-Hecke algebra of type A is a quotient of the BMW algebra. In this thesis, we investigate the structure of these algebras and show they have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. In particular, they are shown to be R-free of rank k^n (2n-1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and Ariki-Koike algebras, one must impose extra so-called "admissibility conditions" on the parameters of the ground ring in order for these results to hold. This is due to potential torsion caused by the polynomial relation of order k imposed on one of the generators of B_n^k. It turns out that the representation theory of B_2^k is crucial in determining these conditions precisely. The representation theory of B_2^k is analysed in detail in a joint preprint with Wilcox in [45] (http://arxiv.org/abs/math/0611518). The admissibility conditions and a universal ground ring with admissible parameters are given explicitly in Chapter 3. The admissibility conditions are also closely related to the existence of a non-degenerate Markov trace function of B_n^k which is then used together with the cyclotomic Brauer algebras in the linear independency arguments contained in Chapter 4. Furthermore, in Chapter 5, we prove the cyclotomic BMW algebras are cellular, in the sense of Graham and Lehrer. The proof uses the cellularity of the Ariki-Koike algebras (Graham-Lehrer [16] and Dipper-James-Mathas [8]) and an appropriate "lifting" of a cellular basis of the Ariki-Koike algebras into B_n^k, which is compatible with a certain anti-involution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by Morton-Wasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and Hauschild-Mosley also attempt similar arguments to establish the freeness and diagram algebra results mentioned above. However, they withdrew their preprints ([14] and [15]), due to issues with their generic ground ring crucial to their linear independence arguments. A similar strategy to that proposed in [14], together with different trace maps and the study of rings with admissible parameters in Chapter 3, is used in establishing linear independency of our basis in Chapter 4. Since the submission of this thesis, new versions of these preprints have been released in which Goodman and Hauschild-Mosley use alternative topological and Jones basic construction theory type arguments to establish freeness of B_n^k and an isomorphism with the cyclotomic Kauffman Tangle algebra. However, they require their ground rings to be an integral domain with parameters satisfying the (slightly stronger) admissibility conditions introduced by Wilcox and the author in [45]. Also, under these conditions, Goodman has obtained cellularity results. Rui and Xu have also obtained freeness and cellularity results when k is odd, and later Rui and Si for general k, under the assumption that \delta is invertible and using another stronger condition called "u-admissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic Nazarov-Wenzl algebras and involve the construction of seminormal representations; their preprints have recently been released on the arXiv. It should also be noted there are slight differences between the definitions of cyclotomic BMW algebras and ground rings used, as explained partly above. Furthermore, Goodman and Rui-Si-Xu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the anti-involution axiom of the original Graham-Lehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.

Let K/Fq be an algebraic function field with full constant field Fq and genus g. Then the divisor class number hK of K/Fq is the order of the quotient group, D0K /P(K), degree zero divisors of K over principal divisors of K. The classification of the function fields K with hK = 1 is done by MacRea, Leitzel, Madan and Queen and the classification of the extensions with class number two is done by Le Brigand. Determination of the necessary and the sufficient conditions for a function field to have class number three is done by H&uml / ulya T&uml / ore. Let k := Fq(T) be the rational function field over the finite field Fq with q elements. For a polynomial N &isin / Fq[T], we construct the Nth cyclotomic function field KN. Cyclotomic function fields were investigated by Carlitz, studied by Hayes, M. Rosen, M. Bilhan and many other mathematicians. Classification of cyclotomic function fields and subfields of cyclotomic function fields with class number one is done by Kida, Murabayashi, Ahn and Jung. Also the classification of function fields with genus one and classification of those with class number two is done by Ahn and Jung. In this thesis, we classified all algebraic function fields and subfields of cyclotomic function fields over finite fields with class number three.