## FIELDS DEFINED BY RADICALS: THEIR TORSION GROUP AND THEIR LATTICE OF SUBFIELDS.

dc.contributor.advisor | Velez, William Yslas | en_US |

dc.contributor.author | ACOSTA DE OROZCO, MARIA TEODORA. | |

dc.creator | ACOSTA DE OROZCO, MARIA TEODORA. | en_US |

dc.date.accessioned | 2011-10-31T16:57:06Z | |

dc.date.available | 2011-10-31T16:57:06Z | |

dc.date.issued | 1987 | en_US |

dc.identifier.uri | http://hdl.handle.net/10150/184040 | |

dc.description.abstract | Let L/F be a finite separable extension. L* = L\{0}, and T(L*/F*) be the torsion subgroup of L*/F*. We explicitly determined T(L*/F*) when L/F is an abelian extension. This information is used to study the structure of T(L*/F*). In particular T(F(α)*/F*) when αᵐ = a ∈ F is explicitly determined. Let Xᵐ - a be irreducible over F with char F χ m and let α be a root of Xᵐ - a. We study the lattice of subfields of F(α)/F and to this end C(F(α)/F,k) is defined to be the number of subfields of F(α) of degree k over F. C(f(α)/F,pⁿ) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) and set n = [N:F], then C(F(α)/F,k) = C(F(α)/F,(k,n)) = C(N/F,(k,n)). The irreducible binomials X⁸ - b, X⁸ - c are said be equivalent if there exist roots β⁸ = b, γ⁸ = c that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. These results are applied the study of normal binomials and those irreducible binomials X²ᵉ - a which are normal over F(charF ≠ 2) together their Galois groups are characterized. We finished by considering the radical extension F(α)/F, αᵐ ∈ F, where the binominal Xᵐ - αᵐ is not necessarily irreducible. We see that in the case not every subfield of F(α)/F is the compositum of subfields of prime power order. We determine some conditions such that if F ⊆ H ⊆ F(α) with [H:F] = pᵘq, p a prime, (p,q) = 1, then there exists a subfield F ⊆ R ⊆ H where [R:F] = pᵘ. | |

dc.language.iso | en | en_US |

dc.publisher | The University of Arizona. | en_US |

dc.rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. | en_US |

dc.subject | Torsion theory (Algebra) | en_US |

dc.subject | Lattice theory. | en_US |

dc.title | FIELDS DEFINED BY RADICALS: THEIR TORSION GROUP AND THEIR LATTICE OF SUBFIELDS. | en_US |

dc.type | text | en_US |

dc.type | Dissertation-Reproduction (electronic) | en_US |

dc.identifier.oclc | 698461896 | en_US |

thesis.degree.grantor | University of Arizona | en_US |

thesis.degree.level | doctoral | en_US |

dc.identifier.proquest | 8712856 | en_US |

thesis.degree.discipline | Mathematics | en_US |

thesis.degree.discipline | Graduate College | en_US |

thesis.degree.name | Ph.D. | en_US |

refterms.dateFOA | 2018-05-29T08:17:41Z | |

html.description.abstract | Let L/F be a finite separable extension. L* = L\{0}, and T(L*/F*) be the torsion subgroup of L*/F*. We explicitly determined T(L*/F*) when L/F is an abelian extension. This information is used to study the structure of T(L*/F*). In particular T(F(α)*/F*) when αᵐ = a ∈ F is explicitly determined. Let Xᵐ - a be irreducible over F with char F χ m and let α be a root of Xᵐ - a. We study the lattice of subfields of F(α)/F and to this end C(F(α)/F,k) is defined to be the number of subfields of F(α) of degree k over F. C(f(α)/F,pⁿ) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) and set n = [N:F], then C(F(α)/F,k) = C(F(α)/F,(k,n)) = C(N/F,(k,n)). The irreducible binomials X⁸ - b, X⁸ - c are said be equivalent if there exist roots β⁸ = b, γ⁸ = c that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. These results are applied the study of normal binomials and those irreducible binomials X²ᵉ - a which are normal over F(charF ≠ 2) together their Galois groups are characterized. We finished by considering the radical extension F(α)/F, αᵐ ∈ F, where the binominal Xᵐ - αᵐ is not necessarily irreducible. We see that in the case not every subfield of F(α)/F is the compositum of subfields of prime power order. We determine some conditions such that if F ⊆ H ⊆ F(α) with [H:F] = pᵘq, p a prime, (p,q) = 1, then there exists a subfield F ⊆ R ⊆ H where [R:F] = pᵘ. |

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