# Module MA2215: Fields, Rings and Modules

**Credit weighting (ECTS)**- 5 credits
**Semester/term taught**- Michaelmas term 2012-13
**Contact Hours**- 11 weeks, 3 lectures including tutorials per week
**Lecturer**- Prof. Vladimir Dotsenko
**Learning Outcomes**-
On successful completion of this module, students will be able to
- State definitions of concepts used in the module, and prove their simple properties;
- Describe rings and fields commonly used in the module, and perform computations in them;
- State theoretical results of the module, 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 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**-
In Modules 1111 and 1214, you encountered algebraic structures such as groups and vector spaces. In this module 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.

**Module Prerequisite**- MA1111 Linear Algebra I, MA1214 Introduction to group theory
**Assessment Detail**-
This module will be examined in a 2 hour
**examination**in Trinity term. C**ontinuous assessment**is via weekly home assignments and it will contribute to the final grade for the module at the annual examination session. The final mark is 80% of the exam mark plus 20% of continuous assessment. The supplemental will consist of 100% examination.