Status: List A

Commitment: 30 lectures + assessment sheets + tests.

Prerequisites: MA106 Linear Algebra, MA245 Algebra II: Groups and Rings.

Content: Galois theory is the study of solutions of polynomial equations. You know how to solve the quadratic equation by completing the square, or by that formula involving plus or minus the square root of the discriminant . 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 has 5 roots , and the corresponding symmetry group is the permutation group 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. The relation between roots and coefficients of a polynomial: elementary symmetric functions; complex roots of unity; and solutions by radicals of cubic and quartic equations.
2. The characteristic of a field and the prime subfield.
3. Factorisation and ideal theory in the polynomial ring ; the structure of a primitive field extension.
4. Field extensions and characterisation of finite normal extensions as splitting fields.
5. The structure and construction of finite fields.
6. Counting field homomorphisms; the Galois group and the Galois correspondence.
7. Radical field extensions.
8. Soluble groups and solubility by radicals of equations.

Books:

Lecture notes will be on sale from the front office, or are available from the lecturer's website

IT Adamson, Introduction to Field Theory, Oliver & Boyd.

E Artin, Galois Theory, University of Notre Dame.

DJH Garling, A course in Galois theory, CUP.

IN Stewart, Galois Theory, Chapman and Hall.

BL van der Waerden, Algebra or Modern algebra, vol 1.

S Lang, Algebra, Springer.

IR Shafarevich, Basic notions of algebra, Springer. (Strongly recommended as an extended essay in philosophical terms on the meaning of algebra - but it won't help much for the exam).

J-P Tignol, Galois' theory of algebraic equations, World Scientific. (Historically aware treatment of all the main issues)

Assessment: 3-hour examination (85%), assessed worksheets and tests (15%).

Lecturer: Martin Bright