Fields and Applications

This is an introduction to the theory of Field Extensions, their Galois groups and the application of finite fields to constructing BCH codes.
Learning Outcomes
1. State the definition of a field and finite extensions of a field.
2. Compute the degree of such an extension.
3. Define the notion of algebraic number, transcendental number, the algebraic closure of a field.
4. State the 3 famous problems of ancient Greek geometry, Ruler and Compass constructions, and how Field Theory contributes to answering these problems and also the construction of regular n-gons via roots of cyclotomic polynomials.
5. Define the notion of automorphisms of an extension field relative to the field of rational numbers.
6. Be able to construct the splitting field of an irreducible polynomial over Q and the corresponding Galois group of automorphisms in the case of small degree.
7. Know why there is no general formula for 'solving the quintic by radicals' and what that expression means.
8. Construct finite fields of small order, properties of finite fields, the Frobenius automorphism. The formula of Gauss for the number of monic irreducible polynomials of degree n over a given finite field.
9. Use finite fields in constructing BCH codes of a designated distance d over such a field.