At the end of the course a succesful student will
- Identify different types of field extensions and state their main properties,
- Find minimal polynomials, test a given polynomial for irreducibility,
- Compute the set of embeddings of a given field into another,
- Carry out certain impossibility proofs, especially using degree of a field extension,
- Find the splitting field of a set of polynomials,
- Determine if a given element or extension is separable,
- Identify algebraically independent elements and compute transcendence degree,
- Find the lattice of intermediate fields between two fields,
- Compute the Galois group of a field extension,
- Match the lattice of intermediate fields between two fields and subgroups of the Galois group through the Galois correspondence maps.