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Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2017.tde-12092017-141837
Document
Author
Full name
David Alberto Saldaña Monteza
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Borges Filho, Herivelto Martins (President)
Mirzaii, Behrooz
Orihuela, Fernando Eduardo Torres
Tizziotti, Guilherme Chaud
 
Title in Portuguese
Fecho Galoisiano de sub-extensões quárticas do corpo de funções racionais sobre corpos finitos
Keywords in Portuguese
Corpo de funções
Corpos finitos
Resolvente cúbica
Teorema de Bézout
Teoria de Galois
Abstract in Portuguese
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.
 
Title in English
Galois closures of quartic sub-fields of rational function fields over finite fields
Keywords in English
Bézout theorem
Cubic resolvent
Finite fields
Function fields
Galois theory
Abstract in English
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.
 
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Publishing Date
2017-09-12
 
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