Module Specification

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.
1. Module Title Commutative Algebra
2. Module Code MATH247
3. Year Session 2023-24
4. Originating Department Mathematical Sciences
5. Faculty Fac of Science & Engineering
6. Semester Second Semester
7. CATS Level Level 5 FHEQ
8. CATS Value 15
9. Member of staff with responsibility for the module
Professor AV Pukhlikov Mathematical Sciences Pukh@liverpool.ac.uk
10. Module Moderator
11. Other Contributing Departments  
12. Other Staff Teaching on this Module
Dr V Guletskii Mathematical Sciences vladimir.guletskii@liverpool.ac.uk
13. Board of Studies
14. Mode of Delivery
15. Location Main Liverpool City Campus
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
16. Study Hours 33

  11

    4

48
17.

Private Study

102
18.

TOTAL HOURS

150
 
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other
19. Timetable (if known)            
 
20. Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

 
21. Modules for which this module is a pre-requisite:

 
22. Co-requisite modules:

 
23. Linked Modules:

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

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

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

27. Aims
 

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

 
28. 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.

 
29. Teaching and Learning Strategies
 

Material is presented during lectures (3 hours per week). Tutorials (1 hour per week) are used for consolidation and practice, and for help with individual questions.

 
30. 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.

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

Assessment

32. EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  final exam This is an anonymous assessment. 120 70
33. CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  Class Test 60 30