Galois theory

TB1, 2011/2012:   Dr. Tim Browning and Dr Andy Booker.

Syllabus

1. Polynomial rings. Irreducible polynomials.
2. Field extensions. Algebraic and transcendental elements. Simple extensions. Degree of an extension. Splitting fields. Algebraic closure.
3. Impossibility of some geometric constructions.
4. Fixed fields and Galois groups. Splitting fields and normal extensions. Separable extensions.
5. The fundamental theorem of Galois theory.
6. Solutions of polynomials by radicals. Insolubility of the general quintic.
7. Finite fields. Transcendental elements and algebraic independence. Applications. The fundamental theorem of algebra.

Reading (among many possible sources)

• D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, 1986.
• Ian Stewart, Galois Theory, 3rd ed. Chapman & Hall, 2003.
• Emil Artin, Galois Theory, New ed. Dover, 1998.

Problem sheets

Problem sheets will be handed out every second monday, due the following tuesday.

• Problem sheet 1
• Problem sheet 2
• Problem sheet 3
• Problem sheet 4
• Problem sheet 5

Past exam papers

• Galois theory exam 2008
• Galois theory exam 2009