Fields, rings, and modules, TCD 2012/13
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.
Syllabus
- 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.
Learning outcomes
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).
Materials
Homework 1 [PDF]Homework 2 [PDF]
Homework 3 [PDF]
Homework 4 [PDF]
Homework 5 [PDF]
Homework 6 [PDF]
Homework 7 [PDF]
Homework 8 [PDF]
Homework 9 [PDF]
Tutorial 1 [PDF]
Tutorial 2 [PDF]
Tutorial 3 [PDF]
Tutorial 4 [PDF]
Solutions to Homework 1 [PDF]
Solutions to Homework 2 [PDF]
Solutions to Homework 3 [PDF]
Solutions to Homework 4 [PDF]
Solutions to Homework 5 [PDF]
Solutions to Homework 6 [PDF]
Solutions to Homework 7 [PDF]
Solutions to Homework 8 [PDF]
Solutions to Homework 9 [PDF]
Solutions to Tutorial 1 [PDF]
Solutions to Tutorial 2 [PDF]
Solutions to Tutorial 3 [PDF]
Solutions to Tutorial 4 [PDF]
Sample exam [PDF]
Textbooks
Two main recommended textbooks are:- Peter J. Cameron, "Introduction to algebra"
- John R. Durbin, "Modern algebra: an introduction"