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 |
Prof AV Pukhlikov Mathematical Sciences Pukh@liverpool.ac.uk |
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| Year | CATS Level | Semester | CATS Value |
| Session 2016-17 | 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. |
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Learning Outcomes |
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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. |
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Syllabus |
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| 1 |
• 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.
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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. Explanation of Reading List: |
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Pre-requisites before taking this module (other modules and/or general educational/academic requirements): |
| MATH101; MATH102; MATH103 *MATH142 or MATH244 are alternative prerequisites for MATH343 |
Co-requisite modules: |
Modules for which this module is a pre-requisite: |
| MATH449; MATH343*; MATH444; MATH448; MATH342 |
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: |
| Programme:G100 Year:2 Programme:G101 Year:2 Programme:G110 Year:2 Programme:G1N2 Year:2 Programme:G1R9 Year:2 Programme:G1X3 Year:2 Programme:GG13 Year:2 Programme:GL11 Year:2 Programme:GN11 Year:2 Programme:GR11 Year:2 Programme:GG15 Year:2 Programme:GV15 Year:2 Programme:G1F7 Year:2 Programme:BCG0 Year:2 Programme:L000 Year:2 Programme:Y001 Year:2 |
Assessment |
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| EXAM | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
| Unseen Written Exam | 2.5 hours | Second semester | 90 | Yes | Assessment 2 Notes (applying to all assessments) 10 pieces of homework will add up to a total 10% of marks This work is not marked anonymously. All answers to Section A and Section B will be taken into account. The marks noted indicate the relative weight of the questions. | |
| CONTINUOUS | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
| Coursework | Weeks 2 to 11 | Second semester | 10 | No reassessment opportunity | Standard UoL penalty applies | Assessment 1 There is no reassessment opportunity, |