Important: On Thursday December 13, only the class at 12am will take place.
In Modules 1111 and 1214, you encountered algebraic structures such as groups and vector spaces. In this course we'll study other algebraic structures that commonly occur. We start by studying rings, which come about when you consider addition and multiplication (but not division) from an abstract point of view. If we throw division into the mix, then we get the definition of a field. We'll look at how one field can be extended to get a larger field, and use this theory to solve some geometric problems that perplexed the Greeks and remained unsolved for 2,000 years. We'll also talk about modules over a ring, which generalise the idea of a vector space over a field.
- Rings; examples, including polynomial rings and matrix rings. Subrings, homomorphisms, ideals, quotients and the isomorphism theorems.
- Integral domains, unique factorisation domains, principal ideal domains, Euclidean domains. Gauss' lemma and Eisenstein's criterion.
- Fields, the field of quotients, field extensions, the tower law, ruler and compass constructions, construction of finite fields.
- Modules over rings: examples.
On successful completion of this module, students will be able to:
- State definitions of concepts used in the course, and prove their simple properties
- Describe rings and fields commonly used in the course, and perform computations in them
- State theoretical results of the course, demonstrate how one can apply them, and outline proofs of some of them (e.g. first isomorphism theorem, or ``an Euclidean domain is a principal ideal domain'', or ``a principal ideal domain is a unique factorisation domain'')
- Perform and apply the Euclidean algorithm in a Euclidean domain
- Give examples of sets where some of the defining properties of fields, rings and modules fail, and give examples of fields, rings and modules satisfying 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 courses 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).
Two main recommended textbooks are:
- Peter J. Cameron, Introduction to algebra
- John R. Durbin, Modern algebra: an introduction
Both of these books cover more topics, but almost every single thing we discuss in class is in at least one of them.
All handouts are downloadable from this page. Filenames should be interpreted as follows:
files called 2215HW*.pdf contain home assignments,
files called 2215A*.pdf contain brief solutions to home assignments (since those are posted right after I collect assignments in class, I cannot accept late assignments),
files called 2215T*.pdf contain tutorial sheets,
files called 2215T*ans.pdf contain solutions to tutorial sheets.
The last home assignment is Assignment 9, the last tutorial is Tutorial 4. There will be no more assignments and/or tutorials for this module.