MTH4150 Algebra 2: Rings and fields

When you finish this unit, you should be able to:

1. 1

   Formulate and critically analyse abstract concepts in algebra.

2. 2

   Apply a wide range of proof techniques to prove complex mathematical results.

3. 3

   Master the manipulation and application of the most common rings and fields: integers, integers modulo n, matrix rings, rationals, real and complex numbers, as well as more general structures such as number fields, algebraic extension fields, splitting fields, algebraic integers, and finite fields.

4. 4

   Exhibit an in-depth understanding of different types of rings, such as integral domains, principal ideal domains, unique factorisation domains, Euclidean domains, fields, and skew-fields, amongst these are the Gaussian integers and quaternions, the most well-known skew-fields.

5. 5

   Demonstrate understanding of the classification of finite fields.

6. 6

   Generalise and apply known concepts over the integers to other domains, for example, use the Euclidean algorithm and factorisation algorithms in the algebra of polynomials.

7. 7

   Construct larger fields from smaller fields (field extensions and splitting fields), demonstrating a deep understanding of these processes.

8. 8

   Apply field theory to coding theory and demonstrate understanding of the classification of cyclic codes.