MAU22102 Abstract algebra II: Fields, rings and modules
| Module Code | MAU22102 |
|---|---|
| Module Title | Abstract algebra II: Fields, rings and modules |
| Semester taught | Semester 2 |
| ECTS Credits | 5 |
| Module Lecturer | Prof. Sergey Mozgovoy |
| Module Prerequisites | MAU22101 Group theory |
Assessment Details
- This module is examined in a 2-hour examination at the end of Semester 2.
- Continuous assessment contributes 15% towards the overall mark.
- Any failed components are reassessed, if necessary, by an exam in the reassessment session.
Contact Hours
11 weeks of teaching with 3 lectures per week.
Learning Outcomes
On successful completion of this module, students will be able to
- State definitions of concepts introduced in the module and prove their main properties.
- Describe fields and rings introduced in the module and perform computations in them.
- State theoretical results covered in the module and outline their proofs.
- Perform and apply the Euclidean algorithm in a Euclidean domain.
- Give examples of sets for which some of the defining properties of fields, rings and modules fail, and give examples of fields, rings and modules which satisfy some additional constraints.
- State and prove the tower law, and use it to prove the impossibility of some classical ruler and compass geometric constructions.
- Identify concepts introduced in other modules as particular cases of fields, rings and modules (e.g. functions on the real line as a ring, abelian groups and vector spaces as modules).
Module Content
- Rings, subrings, homomorphisms, ideals, quotients and isomorphism theorems.
- Integral domains, unique factorisation domains, principal ideal domains, Euclidean domains, Gauss' lemma and Eisenstein's criterion.
- Fields, field of quotients, field extensions, the tower law, ruler and compass constructions, construction of finite fields.
- Modules over rings.