School of Mathematics School of Mathematics
MA2215 - Fields, rings and modules 2011-12 (SF Mathematics
SF Two-Subject Moderatorship )
Lecturer: Prof. E. Vergara Diaz

Requirements/prerequisites: MA1214

Duration: Michaelmas Term (11 weeks)

Number of lectures per week: 3 including tutorials

Assessment: Regular assignments and tutorial work.

ECTS credits: 5

End-of-year Examination: 2 hour end of year examination in Trinity Term

Description:

For more details consult the website: http://www.maths.tcd.ie/~levene/2215

Last year, you met several algebraic structures: groups, fields, vector spaces and sets of matrices. In this course we'll 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

  1. Rings; examples, including polynomial rings and matrix rings. Subrings, homomorphisms, ideals, quotients and the isomorphism theorems.
  2. Integral domains, unique factorisation domains, principal ideal domains, Euclidean domains. Gauss' lemma and Eisenstein's criterion.
  3. Fields, the field of quotients, field extensions, the tower law, ruler and compass constructions, construction of finite fields.
  4. Modules and examples. Direct sum decompositions and applications.

Textbooks:

Learning Outcomes: On successful completion of this module, students will be able to:

Sep 29, 2011


File translated from TEX by TTH, version 2.70.
On 29 Sep 2011, 10:54.