Seja p um primo, considere q = pe com e ≥ 1 inteiro. Dado o polinômio f (x) = x4+ax3+bx2+ cx+d ∈ Fq[x], consideremos o polinômio F(T) = T4 +aT3 +bT2 +cT + d - y ∈ Fq(y)[T], com y = f (x) sobre Fq(y). O objetivo desse trabalho é determinar o número de polinômios f (x) que tem seu grupo de galois associado GF isomorfo a cada subgrupo transitivo (prefixado) de S4. O trabalho foi baseado no artigo: Galois closures of quartic sub-fields of rational function fields, usando equações auxiliares associadas ao polinômio minimal F(T) de graus 3 e 2 (DUMMIT, 1994); bem como uma caraterização das curvas projetivas planas de grau 2 não singulares. Se car(k) ≠ 2, associamos a F(T) sua cúbica resolvente RF(T) e seu discriminante ΔF. Em seguida obtemos condições para GF ≅ C4 (vide Teorema 2.9), que é ocaso fundamental para determinação dos demais casos. Se car(k) = 2, procuramos determinar condições para GRF ≅ A3, associando ao polinômio RF(T) sua quadrática resolvente P(T) (vide a Proposição 2.13). Apos ter homogeneizado P(T), usamos uma das consequências do teorema de Bézout, a saber, uma curva algébrica projetiva plana C de grau 2 é irredutível se, e somente se, C não tem pontos singulares. Nesta dissertação obtemos resultados semelhantes com uma abordagem relativamente diferente daquela usada pelo autor R. Valentini. / Let be p a prime, q = pe whit e ≥ 1 integer. Let a polynomial f (x) = x4+ax3+bx2+cx+d ∈ Fq[x], considering the polynomial F(T)=T4+aT3+bT2+cT +d, with y= f (x) over Fq(y)[T]. The purpose of the current research is to determine the numbers of polynomials f (x) which have its associated Galois group GF, this GF is isomorphic for each transitive subgroup (prefixed) of A4. This project is based on the article: Galois closures of quartic sub-fields of rational function fields, using auxiliary equations associated to the minimal polynomial F(T) of degrees 3 and 2 (DUMMIT, 1994); besides a characterization of non-singular projective plane curves of degree 2 was used. If car(k) ≠ 2, associated to F(T) the resolvent cubic RF(T) and its discriminant ΔF then conditions for GF are obtained as GF ≅ C4 which is the fundamental case for determining the other cases (Theorem 2.9). If car(k) = 2, to find conditions for GRF ≅ A3, associated to the polynomial RF(T) its resolvent quadratic p(T) (Proposition 2.13). Homogenizing p(T), one of the consequences of the Bezout theorem was applied. It is, a projective plane curve C, which grade 2, is irreducible if and only if C is smooth. In the current dissertation, similar results were obtained using a different approach developed by the author R. Valentini.

Seja p um primo, considere q = pe com e ≥ 1 inteiro. Dado o polinômio f (x) = x4+ax3+bx2+ cx+d ∈ Fq[x], consideremos o polinômio F(T) = T4 +aT3 +bT2 +cT + d - y ∈ Fq(y)[T], com y = f (x) sobre Fq(y). O objetivo desse trabalho é determinar o número de polinômios f (x) que tem seu grupo de galois associado GF isomorfo a cada subgrupo transitivo (prefixado) de S4. O trabalho foi baseado no artigo: Galois closures of quartic sub-fields of rational function fields, usando equações auxiliares associadas ao polinômio minimal F(T) de graus 3 e 2 (DUMMIT, 1994); bem como uma caraterização das curvas projetivas planas de grau 2 não singulares. Se car(k) ≠ 2, associamos a F(T) sua cúbica resolvente RF(T) e seu discriminante ΔF. Em seguida obtemos condições para GF ≅ C4 (vide Teorema 2.9), que é ocaso fundamental para determinação dos demais casos. Se car(k) = 2, procuramos determinar condições para GRF ≅ A3, associando ao polinômio RF(T) sua quadrática resolvente P(T) (vide a Proposição 2.13). Apos ter homogeneizado P(T), usamos uma das consequências do teorema de Bézout, a saber, uma curva algébrica projetiva plana C de grau 2 é irredutível se, e somente se, C não tem pontos singulares. Nesta dissertação obtemos resultados semelhantes com uma abordagem relativamente diferente daquela usada pelo autor R. Valentini. / Let be p a prime, q = pe whit e ≥ 1 integer. Let a polynomial f (x) = x4+ax3+bx2+cx+d ∈ Fq[x], considering the polynomial F(T)=T4+aT3+bT2+cT +d, with y= f (x) over Fq(y)[T]. The purpose of the current research is to determine the numbers of polynomials f (x) which have its associated Galois group GF, this GF is isomorphic for each transitive subgroup (prefixed) of A4. This project is based on the article: Galois closures of quartic sub-fields of rational function fields, using auxiliary equations associated to the minimal polynomial F(T) of degrees 3 and 2 (DUMMIT, 1994); besides a characterization of non-singular projective plane curves of degree 2 was used. If car(k) ≠ 2, associated to F(T) the resolvent cubic RF(T) and its discriminant ΔF then conditions for GF are obtained as GF ≅ C4 which is the fundamental case for determining the other cases (Theorem 2.9). If car(k) = 2, to find conditions for GRF ≅ A3, associated to the polynomial RF(T) its resolvent quadratic p(T) (Proposition 2.13). Homogenizing p(T), one of the consequences of the Bezout theorem was applied. It is, a projective plane curve C, which grade 2, is irreducible if and only if C is smooth. In the current dissertation, similar results were obtained using a different approach developed by the author R. Valentini.

This thesis examines the connections between Bézout domains and elementary divisor domains. I establish what both of these domains are, and I provide some clarifying examples of each. I state and prove some key results that have been established already in the literature. I describe a process by which I tried to show a distinction between Bézout domains and elementary divisor domains, and then provide an explicit example which shows that this process as formulated would not lead to an example of a Bézout domain which is not an elementary divisor domain. Throughout the thesis, I also state open questions that could lead to future research in this area.

In this thesis we study the computation of the greatest common divisor of two polynomials. Firstly, properties of Sylvester matrices are considered as well as their role in computation. We then note, that this approach can be naturally generalized for several polynomials. In the penultimate section, Bézout matrices are studied as an analogy to the Sylvester ones, providing necessary comparison. Extension for more than polynomials is presented here as well. Algorithms corresponding to the individual approaches are presented as well. Finally, the algorithms are implemented in MATLAB and are compared in numerical experiments. Powered by TCPDF (www.tcpdf.org)

In this bachelor's thesis we will enter the world of projective geometry and algebraic geometry. The main part of this thesis focuses on describing projective spaces and their properties. We use the gained knowledge of projective spaces to then study projective varieties, algebraic geometry and Bézouts theorem.

Dans cette thèse, nous visons l'amélioration de quelques algorithmes en algèbre matricielle rapide et plus spécifiquement les algorithmes rapides sur les matrices structurées en calcul formel et numérique. Nous nous intéressons en particulier aux matrices de Hankel et de Toeplitz. Nous introduisons un nouvel algorithme de diagonalisation par blocs approchée de matrices réelles de Hankel. Nous décrivons la relation naturelle entre l'algorithme d'Euclide et notre factorisation par blocs approchée pour les matrices de Hankel associées à deux polynômes, ainsi que pour les matrices de Bézout associées aux mêmes polynômes. Enfin, dans le cas complexe, nous présentons un algorithme révisé de notre diagonalisation par blocs approchée des matrices de Hankel, en calculant la suite des restes et la suite des quotients apparues au cours de l'exécution de l'algorithme d'Euclide.