This project centers around finding specific examples of quintic polynomials that were and were not solvable. This helped to devise a method for finding examples of solvable and unsolvable quintics.

In Part I of the thesis an account is given of the basic algebra of extension fields which is required for the understanding of Galois theory. The fundamental theorem states the relationships of the subgroups of a permutation group of the root field of an equation to the subfields which are left invariant by these subgroups. Extensions of the basic theorem conclude Part I. In part II the solvability of equations by radicals is discussed, for fields of characteristic zero. A discussion of finite fields and primitive roots leads to a criterion for the solvability by radicals of equations over fields of prime characteristic. Finally, a method for determining the Galois group of any equation is discussed. Most of the material in the introductory chapters is taken from Artin's: Galois Theory (cf. p. 120). / Thesis / Master of Arts (MA)

Satisfactory systems of confounding for symmetrical factorial experiments can be constructed oy the familiar methods, using the. theory of Galois fields. Although these methods can be extended to asymmetrical factorial experiments· (White and Hultquist, 1965; Raktoe, 1969) the actual construction of designs becomes much mor:e complicated for the general case and does not always lead to satisfactory plans. A different approach to this problem is to consider balanced factorial experiments (BFE), due to Shah (1958, 1960). Such BFE have a one-to-one relationship to EGD-PBIB designs given by Hinkelmann (1964). The problem of constructing BFE is then equivalent to constructing EGD-PBIB designs. A new method is proposed here to construct such designs. This method is based upon the so-called (1,1, ...,1)th-associate matrix and the operations symbolic direct product (SDP), generalized symbolic direct product (GSDP), symbolic direct multiplication (SDM), and generalized symbolic direct multiplication (GSDM). Let A₁ , A₂, ... , A_n be n factors in a factorial experiment, with A_i having t_i levels (i = 1, 2, ... , n). It is shown that an EGD-PBIB design with blocks of size t_i can be constructed, provided that t_iᵢ ≠ max ( t₁ , t₂, . . . , t_n ). This method is more general and more flexible than the method of Aggarwal (1974) in that any two treatment combinations can be γ-th associates where γ has at least two unity components, and it can be shown the number of possible candidates for such is 2^{n-i l} -1 for blocks of size t_i (i = 1, 2, .. , n -1), where t₁ < t₂ <...< t_n. This method is also more general than the Kronecker product method due to Vartak (1955}. Two types of PBIB designs· are used for reducing the numbers of associa,te classes in EGD-PBIB designs. When the t_i (i = 1, 2, ... , n) are equal, then some EGD-PBIB designs can be reduced to a hypercubic design. The EGD-PBIB designs with block size π [below jεA] t_j, where A is an arbitrary subset of the set {1, 2, ... , n} can be reduced to newly introduced F_A⁽ⁿ⁾-type PBIR designs. Since BFE results very often in designs with a large number of blocks, the notion of partial balanced factorial experiment (PBFE) has been introduced. It is investigated how such designs can be constructed and related to PBIB-designs similar to that between BFE and EGD-PBIB designs. Two new types of PBIB designs have been introduced in this context. / Ph. D.

Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q. / Master of Science

This work explores the unsolvability of the general quintic equation through the lens of Galois theory. We begin by providing a historical perspective on the problem. This starts with the solution of the general cubic equation derived by Italian mathematicians. We then move on to Lagrange's insights on the importance of studying the permutations of roots. Finally, we discuss the critical contributions of Évariste Galois, who connected the solvability of polynomials to the properties of permutation groups. Central to our thesis is the introduction and motivation of key concepts such as fields, solvable groups, Galois groups, Galois extensions, and radical extensions. We rigorously develop the theory that connects the solvability of a polynomial to the solvability of its Galois group. After developing this theoretical framework, we go on to show that there exist quintic polynomials with Galois groups that are isomorphic to the symmetric group S5. Given that S5 is not a solvable group, we establish that the general quintic polynomial is not solvable by radicals. Our work aims to provide a comprehensive and intuitive understanding of the deep connections between polynomial equations and abstract algebra.

This thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.

Neste trabalho, apresentamos um estudo sobre a existência ou não de bases normais auto-duais para extensões galoisianas finitas de corpos, mostrando que toda extensão galoisiana finita de grau ímpar posui uma base normal auto-dual, enquanto que para extensões galoisianas de grau par, apresentamos algumas condições suficientes que garantem a não existência de bases normais auto-duais / In this work, we present a study about the existence or not of self-dual normal bases for finite galoisian extensions of fields, showing that all the odd degree finite galoisian extension has a self-dual normal base, whereas for even degree galoisian extensions, we present some sufficient conditions that assure the non-existence of self-dual normal bases

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.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2006. / This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.