Module Details
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title Commutative Algebra
Code MATH247
Coordinator Professor AV Pukhlikov
Mathematical Sciences
Pukh@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2020-21 Level 5 FHEQ Second Semester 15

Aims

To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory.

Learning Outcomes

(LO1) After completing the module students should be able to: • Work confidently with the basic tools of algebra (sets, maps, binary operations and equivalence relations). • Recognise abelian groups, different kinds of rings (integral, Euclidean, principal ideal and unique factorisation domains) and fields. • Find greatest common divisors using the Euclidean algorithm in Euclidean domains. • Apply commutative algebra to solve simple number-theoretic problems.

Syllabus
 

• Integers, Gaussian integers and polynomials.

• Abelian groups and applications to number theory, e.g. the Chinese remainder theorem.

• Rings. Unique factorization domains. Ideals. Direct sums. Primes and irreducibles.

• Fields. Algebraic extensions. Fields of rational functions.

• Modules. Determinants. The Cayley-Hamilton theorem.

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH101 Calculus I; MATH102 CALCULUS II; MATH103 MATH103 - Introduction to Linear Algebra 

Co-requisite modules:
 

Modules for which this module is a pre-requisite:
 

Programme(s) (including Year of Study) to which this module is available on a required basis:
 

Programme(s) (including Year of Study) to which this module is available on an optional basis:
 

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Final Assessment Open book and remote This is an anonymous assessment. Assessment Schedule (When) :Second semester  1 hour time on task    50       
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Homework 2 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When): Second semester  equivalent to 2-5 si    10       
Homework 4 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When): Second semester  equivalent to 2-5 si    10       
Homework 3 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When): Second semester  equivalent to 2-5 si    10       
Homework 1 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester  Equivalent to 2-5 si    10       
Homework 5 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When): Second semester  equivalent to 2-5 si    10