MTH3150 Algebra 2: Rings and fields

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

1. 1

   Formulate abstract concepts in algebra;

2. 2

   Use a variety of proof-techniques to prove mathematical results;

3. 3

   Work with the most commonly occurring rings and fields: integers, integers modulo n, matrix rings, rationals, real and complex numbers, more general structures such as number fields and algebraic extension fields, splitting fields, algebraic integers and finite fields;

4. 4

   Demonstrate understanding of different types of rings, such as integral domains, principal ideal domains, unique factorisation domains, Euclidean domains, fields, skew-fields; amongst these are the Gaussian integers and the quaternions - the best-known skew field;

5. 5

   Demonstrate understanding of the classification of finite fields;

6. 6

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

7. 7

   Construct larger fields from smaller fields (field extensions and splitting fields);

8. 8

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