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An Introduction to Galois Theory ; Algebra

By Baker, Andrew

Book Id:WPLBN0003761470 Format Type:PDF eBook : File Size: Reproduction Date:2015

Baker, A. (2013). An Introduction to Galois Theory ; Algebra. Retrieved from http://hawaiilibrary.net/

Description
Description: Much of early algebra centred around the search for explicit formulae for roots of polynomial equations in one or more unknowns. The solution of linear and quadratic equations in a single unknown was well understood in antiquity, while formulae for the roots of general real cubics and quartics was solved by the 16th century. These solutions involved complex numbers rather than just real numbers. By the early 19th century no general solution of a general polynomial equation ‘by radicals’ (i.e., by repeatedly taking n-th roots for various n) was found despite considerable effort by many outstanding mathematicians. Eventually, the work of Abel and Galois led to a satisfactory framework for fully understanding this problem and the realization that the general polynomial equation of degree at least 5 could not always be solved by radicals. At a more profound level, the algebraic structure of Galois extensions is mirrored in the subgroups of their Galois groups, which allows the application of group theoretic ideas to the study of fields. This Galois Correspondence is a powerful idea which can be generalized to apply to such diverse topics as ring theory, algebraic number theory, algebraic geometry, differential equations and algebraic topology. Because of this, Galois theory in its many manifestations is a central topic in modern mathematics. In this course we will focus on the following topics. • The solution of polynomial equations over a field, including relationships between roots, methods of solutions and location of roots. • The structure of finite and algebraic extensions of fields and their automorphisms. We will study these in detail, building up a theory of algebraic extensions of fields and their automorphism groups and applying it to solve questions about roots of polynomial equations. The techniques we will meet can also be applied to study the following some of which may be met by people studying more advanced courses. • Classic topics such as squaring the circle, duplication of the cube, constructible numbers and constructible polygons. • Applications of Galois theoretic ideas in Number Theory, the study of differential equations and Algebraic Geometry. There are many good introductory books on Galois Theory, some of which are listed in the Bibliography. In particular, [2, 3, 8] are all excellent sources and have many similarities to the present approach to the material.

Table of Contents
TOC: Integral domains, fields and polynomial rings - Fields and their extensions - Algebraic extensions of fields - Galois extensions and the Galois Correspondence - Galois extensions for fields of positive characteristic - A Galois Miscellany.