Topics:
The course consists of two parts,
Group Theory
and
Field/Galois Theory.
Topics in Group Theory include:
- group actions on sets and permutation representations
- the Sylow theorems
- the Jordan-Hölder theorem
- simplicity of the alternating group An
for n at least 5
- nilpotent groups and solvable groups
Topics in Field/Galois theory include:
- field extensions
- splitting fields, algebraic closures
- separable and inseparable extensions
- cyclotomic polymomials
- finite fields
- the Galois correspondence
- solving equations by radicals, insolvability of the quintic
We will cover roughly Chapters 4,6,13,14 of Dummit-Foote. (Parts of Chapter 3,5
may also be covered.)
You should be
already familiar with the topics in Chapter 1-3, the structure
theorem for finitely generated abelian groups (5.2). You also should
have a sturdy knowledge of linear algebra.
Other resources are:
- Lang, Algebra.
- Artin, Algebra.
- MacLane, Birkhoff, Algebra.
- Van der Waerden, Modern Algebra, vol I.
- Hungerford, Algebra.
- Jacobson, Basic Algebra.