Math 672: Algebra 2.
Desired Learning Outcomes
Math 671 is a prerequisite for this course. In particular, students should have a deep knowledge of groups, rings, and modules. Students should be able to prove difficult theorems on their own.
Minimal learning outcomes
Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.
- Linear Algebra
- Basic theory (including infinite dimensional spaces)
- Dual vector spaces
- Modules over PID's
- Fundamental Theorem
- Applications to Rational and Jordan Canonical Forms of matrices
- Galois Theory
- Field Extensions
- Algebraic, separable, and inseparable extensions
- Splitting fields
- Algebraic closures
- Cyclotomic extensions
- Fundamental Theorem of Galois theory
- Finite fields and their Galois groups
- Composite extensions
- Simple extensions
- Kronecker-Weber Theorem (Statement)
- Galois groups of polynomials
- Transcendental extensions
- Infinite Galois groups
Possible textbooks for this course include (but are not limited to):
If students in the class have not seen module theory, the module theory in Math 671 should be considered a requirement. As time permits, additional topics may be covered. These may include Group representation theory, commutative algebra, homological algebra, lie algebras, or other topics chosen by the instructor.