This thesis is concerned with the Regular Inverse Galois Problem for p-groups over fields of characteristic unequal to p. Building upon results of Saltman, Dentzer characterized a class of finite groups that are automatically realized over every field, and proceeded to show that every group of order dividing p⁴ belongs to this class. We extend this result to include groups of order p⁵, provided that the base field k contains the p³-th roots of unity. The proof involves reducing to certain Brauer embedding problems defined over the rational function field k(x). Through explicit computation, we describe the cohomological obstructions to these embedding problems. Then by applying results about the Brauer group of a Dedekind domain, we show that they all possess solutions. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

This paper is aimed to study the divisibility in integrality domain. So, it was structured that way: at first it is done a basic approach which will be a pre domain for its development, right after it will be done a study about Euclidean domain and Gaussian integer domains culminating in the enforcement of the results gotten in the characterization of prime ideals Gaussian s integers ring. / Este trabalho tem como objetivo estudar a divisibilidade em domínios de integridade para tanto, ele foi estruturado da seguinte forma: inicialmente faz-se uma abordagem básica que servirá como pré-requisito para o seu desenvolvimento, em seguida, faremos um estudo sobre os domínios euclidianos e o domínio dos inteiros de Gauss, culminando com a aplicação dos resultados obtidos na caracterização dos ideais primos do anel dos inteiros de Gauss.

Orientador: Plamen Emilov Koshlukov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-17T15:30:47Z (GMT). No. of bitstreams: 1 Machado_GustavoGrings_M.pdf: 1211967 bytes, checksum: 772eb43184b0ff273c48ec3a47e9ec93 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho estudamos álgebras com identidades polinomiais, focando-se no estudo de álgebras associativas unitárias finitamente geradas. Nosso objetivo é fazer uma demonstração alternativa da não PI-equivalência de álgebras utilizando um invariante conhecido como dimensão de Gelfand-Kirillov. Este invariante tem ganhado importância ultimamente, uma vez que ele é relativamente fácil de calcular e, de certa forma, é capaz de diferenciar o modo com que duas álgebras crescem. Começamos com as definições e resultados básicos de álgebras, álgebras graduadas, identidades polinomiais (graduadas), reduções de identidades polinomiais, etc. Em seguida apresentamos alguns resultados de álgebras com identidades polinomiais finitamente geradas, que permitem uma melhor compreensão dos conceitos de altura e de dimensão de Gelfand-Kirillov. Depois estudamos o Teorema do Produto Tensorial de Kemer (TPT), donde se conclui a PI-equivalência (multilinear) envolvendo álgebras importantes na teoria de PI-álgebras, as álgebras T-primas. Em particular, conclui-se a PI-equivalência sobre corpos de característica zero de M1;1(E) e EE, em que E é a álgebra de Grassmann de um espaço vetorial de base enumerável. Enfim, finalizamos mostrando a não PI-equivalência sobre corpos infinitos de característica positiva maior que dois de M1;1(E) e E E, utilizando-se da dimensão de Gelfand-Kirillov / Abstract: In this work we study algebras with polynomial identities, focusing on the study of finitely generated unitary associative algebras. Our goal is to give an alternative proof of non PI-equivalence of algebras using an invariant known as Gelfand-Kirillov dimension. This invariant has gained importance lately since in many cases it is relatively easy to calculate and, surprisingly, it is able to differentiate the growth of two algebras. We begin with definitions and basic results of algebras, graded algebras, (graded) polynomial identities, reduction of polynomial identities, etc. Afterwards we present some results concerning finitely generated algebras with polynomial identities, which give a better comprehension of the notions of height and Gelfand-Kirillov dimension. Later on we study the Kemer's Tensor Product Theorem (TPT), from which we conclude (multilinear) PI-equivalence involving important algebras in PI-theory, the so called T-prime algebras. In particular, we deduce the PI-equivalence of M1;1(E) and E E over fields of characteristic zero, where E is the infinite dimensional Grassman algebra. Finally, we prove the non PI-equivalence of M1;1(E) and E E over infinite fields of prime characteristic greater than two by means of Gelfand-Kirillov dimension / Mestrado / Algebra / Mestre em Matemática

Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisticas e Computação Cientifica / Made available in DSpace on 2018-08-07T19:30:45Z (GMT). No. of bitstreams: 1 Alves_SergioMota_D.pdf: 708263 bytes, checksum: 1d9ec0b24db06ea81853a6e7d6794b18 (MD5) Previous issue date: 2006 / Resumo: As álgebras verbalmente primas são bem conhecidas em característica 0, já sobre corpos de característica p > 2 pouco sabemos sobre elas. Nesse trabalho vamos discutir algumas diferenças entre estes dois casos de característica sobre corpos infinitos. Iniciamos mostrando que o Teorema do Produto Tensorial de Kemer e duas de suas conseqüências não podem ser transportados para corpos infinitos de característica positiva p > 2. Em seguida, discutiremos algumas propriedades envolvendo as álgebras Aa;b, a saber, mostraremos que as álgebras Aa;b e Ma+b(E) não são PI-equivalentes e que as álgebras Aa;a e Ma;a (E) ­ não são PI-equivalentes, e apresentaremos um resultado que enfatiza a importância dos monômios na determinação do ideal das identidades das álgebras Zn £ Z2-graduadas Aa;b em característica positiva. Por ¯m, apresentaremos modelos genéricos e calcularemos a dimensão de Gelfand-Kirillov para as álgebras relativamente livres de posto m nas variedades determinadas pelas álgebras E ­ E, Aa;b e Ma;a(E) ­ E. Como conseqüência, obteremos a prova da não PI- equivalência entre álgebras importantes para PI-teoria em característica positiva / Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of characteristic p > 2 little is known about them. In this work we discuss some sharp di®erences between these two cases for the characteristic. First we show that the so-called Kemer's Tensor Product Theorem and two of its consequences cannot be extended for infnite fields of positive characteristic p > 2. Afterwards we prove that the algebras Aa;b and Ma+b(E) are not PI equivalent, while the algebras Aa;a and Ma;a(E) ­ E are PI equivalent. Moreover we obtain a result showing the importance of the monomials in the Zn £ Z2-graded T-ideal of the algebra Aa;b. Finally, we exhibit constructions of generic models. By using these models we compute the Gelfand-Kirillov dimension of the relatively free algebras of rank m in the varieties generated by E ­E, Aa;b, and Ma;a(E)­E. As consequence we obtain the PI non equivalence of important algebras for the PI theory in positive characteristic / Doutorado / Algebra / Doutor em Matemática

Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T13:33:07Z (GMT). No. of bitstreams: 1 Fidelis_Marcello_D.pdf: 592299 bytes, checksum: dea5983279c32bbe6e1ffd7e1372fcf4 (MD5) Previous issue date: 2005 / Resumo: Neste trabalho estudamos os produtos tensoriais de T-ideais T-primos sobre corpos infinitos. O comportamento destes produtos tensoriais sobre corpos de caracteristica zero foi descrito por Kemer. Primeiramente mostramos, usando os m'etodos introduzidos por Regev, que tal descri¸cao vale se nos restringirmos apenas aos polinomios multilineares. Num segundo momento, aplicando identidades graduadas, mostramos que o Teorema sobre o Produto Tensorial 'e falso para os T-ideais das 'algebras M1,1(E) e E E, onde E 'e a 'algebra de Grassmann com dimensao infinita; M1,1(E) consiste das matrizes 2 × 2 sobre E tendo somente elementos pares (i.e. centrais) de E na diagonal principal, e a outra diagonal consistindo de elementos 'impares (anticomutitativos) de E. Entao voltamos nossa atencao para outros produtos tensoriais e estudamos suas respectivas identidades graduadas. Obtivemos novas demonstracoes de alguns dos casos do Teorema sobre o Produto Tensorial de Kemer. Note que estas demonstracoes nao dependem da teoria sobre a estrutura dos T-ideais, mas sao "elementares". Finalmente, usando outra vez identidades polinomiais graduadas, mostramos que o Teorema sobre o Produto Tensorial nao 'e valido em mais um caso: quando o corpo base possui caracteristica positiva. Isto vem para mostrar novamente que a teoria sobre a estrutura dos T-ideais e, essencialmente, uma teoria sobre identidades polinomiais multilineares. / Abstract: In this work we study tensor products of T-prime T-ideals over infinite fields. The behaviour of these tensor products over a field of characteristic zero was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the Tensor Product Theorem fails for the T-ideals of the algebras M1,1(E) and E E where E is the infinite dimensional Grassmann algebra; M1,1(E) consists of the 2×2 matrices over E having even (i.e. central) elements of E in the main diagonal, and the other diagonal consisting of odd (anticommuting) elements of E. Then we pass to other tensor products and study the respective graded identities. We obtain new proofs of some cases of Kemer's Tensor Product Theorem. Note that these proofs do not depend on the structure theory of T-ideals but are "elementary" ones. Finally, using graded polynomial identities once again, we show that the Tensor Product Theorem fails in one more case when the base field is of positive characteristic. All this comes to show once more that the structure theory of T-ideals is essentially about the multilinear polynomial identities / Doutorado / Matematica / Doutor em Matemática

No description available.

The ring of real-valued continuous functions on a completely regular frame L is denoted by RL. As usual, βL denotes the Stone-Cech compactification of ˇ L. In the thesis we study ideals of RL induced by sublocales of βL. We revisit the notion of purity in this ring and use it to characterize basically disconnected frames. The socle of the ring RL is characterized as an ideal induced by the sublocale of βL which is the join of all nowhere dense sublocales of βL. A localic map f : L → M induces a ring homomorphism Rh: RM → RL by composition, where h: M → L is the left adjoint of f. We explore how the sublocale-induced ideals travel along the ring homomorphism Rh, to and fro, via expansion and contraction, respectively. The socle of a ring is the sum of its minimal ideals. In the literature, the socle of RL has been characterized in terms of atoms. Since atoms do not always exist in frames, it is better to express the socle in terms of entities that exist in every frame. In the thesis we characterize the socle as one of the types of ideals induced by sublocales. A classical operator invented by Gillman, Henriksen and Jerison in 1954 is used to create a homomorphism of quantales. The frames in which every cozero element is complemented (they are called P-frames) are characterized in terms of some properties of this quantale homomorphism. Also characterized within the category of quantales are localic analogues of the continuous maps of R.G. Woods that characterize normality in the category of Tychonoff spaces. / Mathematical Sciences / Ph. D. (Mathematics)

We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals and d-ideals of the ring RL of continuous real-valued functions on a completely regular frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated to be atly projectable if and only if the ring RL is feebly Baer. On the other hand, the frame of d-ideals is projectable precisely when the frame is cozero-complemented. These ideals give rise to two functors as follows: Sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. We show that, for a certain collection of frame maps, the functor associated with z-ideals preserves and re ects the property of having a left adjoint. A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal prime ideals it contains. In the penultimate chapter we give several characterisations for the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra, then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example is provided to show that the converse fails. Finally, piggybacking on results in classical rings of continuous functions, we show that, exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition that every reduced f-ring with bounded inversion is the ring of fractions of its bounded part relative to those elements in the bounded part which are units in the bigger ring. We close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L. / Mathematical Sciences / Mathematics / D.Phil. (Mathematics)