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

Justify with reasoned logical argument basic properties of
Noetherian modules and finite field extensions.

Justify with reasoned logical argument results concerning
the structure of finitely-generated modules over integral domains
and principal ideal domains.

Describe and justify with reasoned logical argument basic
properties of algebraic integers.

Module Content

The module will cover the following topics:

Factorization in Integral Domains. Principal Ideal Domains.

Basic properties of modules over unital commutative rings.
Noetherian modules. Noetherian rings. Hilbert's Basis Theorem.

Linear independence and free modules.
Free modules over integral domains. Torsion modules.
Free modules of finite rank over principal ideal domains.
Torsion-free modules.
The classification theorem for finitely-generated modules
over principal ideal domains.
The Jordan Normal Form.

Algebraic numbers and algebraic integers.

Lecture notes, problems, worked solutions to selected past
examination questions and further information relevant
to the module are available from the module webpage at
http://www.maths.tcd.ie/~dwilkins/Courses/MA3412/.

Module Prerequisite

Either Fields, Rings and Modules (MA2215) or
Abstract Algebra I (MA3411).

Assessment Detail

This module will be examined in a 2-hour examination in Trinity term.
There is no continuous assessment.