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M4071 Galois theory

Aims: The aim of this unit is to study fields and their symmetry, and thereby understand some classical problems of geometry and algebra. These problems can all be translated into polynomial equations, and Galois discovered that the solution of such equations depends on their symmetry. He also recognised that symmetry is captured by the group concept, and found a property of groups (now called "solvability") which detects solvability of equations.
Galois theory develops these ideas systematically to explain why a variety of problems cannot be solved. In the process, some important concepts of group theory are developed and applied.

Syllabus: Formulation of classical problems in field theory. The arithmetic of polynomials. Irreducibility. Unsolvability of the classical ruler and compass problems. Automorphisms of fields and the Galois group. Radical extensions. Unsolvability of the general quintic equation.

Prerequisites: Please see lecturer.

Recommended texts:
Fraleigh, J. B., A first course in abstract algebra, Wiley, 1989.
Petsinis, T., The French mathematician, Penguin, 1997.
Stillwell, J. C., Elements of algebra, Springer-Verlag, 1994.

Lecturer: Tom Hall



 
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