Galois theory gives a relation between field extensions on the one side and subgroups of a certain automorphism group on the other. In this course we shall develop the theory of vector spaces and polynomials over arbitrary fields and use them to study fields and their extensions. Ultimately we link this up with the automorphism group of a field culminating with the famous Fundamental Theorem of Galois Theory. Along the way we shall see some applications to the problem of constructibility by ruler and compass and to the solvability of equations using only radicals.
Rough lecture plan
Week 1: Fields, vector spaces over arbitrary fields, extension fields. Week 2: Polynomials over arbitrary fields. Week 3: Algebraic extensions and constructibility. Week 4: Galois theory. Week 5: Solving equations by radicals. Week 6: Further examples and applications.
A successful student should be able to do the following by the end of this course:
- Work comfortably with arbitrary fields
- Work with vector spaces over arbitrary fields
- Understand which theorems from linear algebra hold over arbitrary fields and which don't
- Work with polynomials over arbitrary fields
- Understand the relation between polynomials and field extensions
- Understand and work with the relation between subgroups of Gal(E/F) and intermediate fields between E and F.
- Calculate Galois groups of some simple polynomials