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 |
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Year | CATS Level | Semester | CATS Value |
Session 2024-25 | Level 5 FHEQ | Second Semester | 15 |
Aims |
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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 |
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(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 |
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• 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 |
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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; MATH103 Introduction to Linear Algebra; MATH102 CALCULUS II |
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 |
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EXAM | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
final exam This is an anonymous assessment. | 120 | 70 | ||||
CONTINUOUS | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
Class Test | 60 | 30 |