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MA3D5 Galois Theory

Lecturer: Gavin Brown

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 lectures + assessment sheets

Assessment: 3-hour examination (85%), best 3 out of 4 assessed worksheets (15%).

Prerequisites: MA106 Linear Algebra, MA249 Algebra II: Groups and Rings

Leads To:

Content: Galois theory is the study of solutions of polynomial equations. You know how to solve the quadratic equation $ ax^2+bx+c=0 $ by completing the square, or by that formula involving plus or minus the square root of the discriminant $ b^2-4ac $ . The cubic and quartic equations were solved ``by radicals'' in Renaissance Italy. In contrast, Ruffini, Abel and Galois discovered around 1800 that there is no such solution of the general quintic. Although the problem originates in explicit manipulations of polynomials, the modern treatment is in terms of field extensions and groups of ``symmetries'' of fields. For example, a general quintic polynomial over $Q$ has five roots $ \alpha_1.\dots.\alpha_5 $ , and the corresponding symmetry group is the permutation group $ S_5 $ on these.

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.

Objectives: By the end of the module the student should understand

1. Solution by radicals of cubic equations and (briefly) of quartic equations.
2. The characteristic of a field and its prime subfield. Field extensions as vector spaces.
3. Factorisation and ideal theory in the polynomial ring k[x]; the structure of a simple field extension.
4. The impossibility of trisecting an angle with straight-edge and compass.
5. The existence and uniqueness of splitting fields.
6. Groups of field automorphisms; the Galois group and the Galois correspondence.
7. Radical field extensions; soluble groups and solubility by radicals of equations.
8. The structure and construction of finite fields.

Books: DJH Garling, A course in Galois theory, CUP.

IN Stewart, Galois Theory, Chapman and Hall.

Additional Resources

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