## Math 405: Groups, Rings and Fields

Study Guide for Exam 1

The first midterm exam consists of two parts. The first part will be held in class on **Monday, October 16**, during our regular class time. This is a closed book, 50 minute exam. The second part will be a take-home portion, which you will receive after completing the in-class part. You may use class notes and the textbooks on reserve for the take-home component, but no other outside resources are permitted. This part will be due at the beginning of class on **Friday, October 20**.

The following is a list of topics intended to help you organize
the material we have covered in class as you study for your exam. It is only
intended to serve as a guideline, and may not explicitly mention everything that
you need to study.

Know definitions, examples and associated theorems for the following concepts:

- Lie algebras
- bilinearity
- antisymmetry
- Jacobi identity
- subalgebras
- general/special linear algebra
- commutator
- derivations
- abelian

- Lie algebra homomorphisms
- isomorphisms
- representation
- adjoint map
- outer and inner derivations
- ideals and quotients
- simple Lie algebras

- Solvability and nilpotence
- Semisimplicity
- Engel's Theorem and Lie's Theorem
- Cartan's criterion
- Killing forms

Maintained by ynaqvi
and last modified 10/09/17