To present an introduction to Galois theory in the context of arbitrary field extensions and apply it to a number of historically important mathematical problems. Consider a field, such as the rational numbers, and consider a larger field containing it, which can also be thought of as a vector space over the original field. One question that can be asked is: what symmetries or automorphisms of the bigger field exist that act as the identity on the smaller field? The Galois Correspondence connects the answer to this question with the properties of a group of permutations of the roots of a polynomial. This relationship can be used in different directions to translate a problem from one part of algebra to another part where it may be easier to solve.
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Lecturer: Adam Thomas. Content : Galois theory is the study of solutions of polynomial equations. In contrast, Ruffini, Abel and Galois discovered around that there is no such solution of the general quintic. Aims : The course will discuss the problem of solutions of polynomial equations both in explicit terms and in terms of abstract algebraic structures. The course demonstrates the tools of abstract algebra linear algebra, group theory, rings and ideals as applied to a meaningful problem.
Solution by radicals of cubic equations and briefly of quartic equations. The characteristic of a field and its prime subfield. Field extensions as vector spaces. Factorisation and ideal theory in the polynomial ring k[x]; the structure of a simple field extension. The impossibility of trisecting an angle with straight-edge and compass.
The existence and uniqueness of splitting fields. Groups of field automorphisms; the Galois group and the Galois correspondence. Radical field extensions; soluble groups and solubility by radicals of equations. The structure and construction of finite fields. Archived Pages: Pre Year 3 regs and modules G G Year 4 regs and modules G Past Exams Core module averages.
Objectives : By the end of the module the student should understand 1. Additional Resources Archived Pages: Pre
B3.1 Galois Theory (2016-2017)
Note: This is an old module occurrence. You may wish to visit the module list for information on current teaching. In the cases of interest, this is a finite group, and there is a tight link called the Galois correspondence between the structure of G and the subfields of K. If K is generated over the rationals by the roots of a polynomial f x , then G can be identified as a group of permutations of the set of roots. One can then use the Galois correspondence to help find formulae for the roots, generalising the standard formula for the roots of a quadratic. It turns out that this works whenever the degree of f x is less than five.
Rings and Modules is essential and Group Theory is recommended. Students who have not taken Part A Number Theory should read about quadratic residues in, for example, the appendix to Stewart and Tall. This will help with the examples. The course starts with a review of second-year ring theory with a particular emphasis on polynomial rings, and a discussion of general integral domains and fields of fractions. This is followed by the classical theory of Galois field extensions, culminating in some of the classical theorems in the subject: the insolubility of the general quintic and impossibility of certain ruler and compass constructions considered by the Ancient Greeks.