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## ECM2711 - Groups, Rings and Fields (2015)

MODULE TITLE | Groups, Rings and Fields | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM2711 | MODULE CONVENER | Dr Henri Johnston (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 | 0 | 0 |

Number of Students Taking Module (anticipated) | 79 |
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Building on the foundations established in ECM1706 or equivalent, you will learn about the key unifying structures that form a basis for any further modules in Algebra, Number Theory and Geometry at Stages 3 and 4

This module gives you the opportunity to further refine your skills in problem-solving, axiomatic reasoning and the formulation of mathematical proofs.

Students that take this course are expected to have a strong interest in Algebra. The level of difficulty of this course is considerably higher than that of Numbers, Symmetries and Groups (ECM1706).

Prerequisite modules: ECM1706 and ECM1701 or equivalent

The aim of this module is to give you an introduction to the theory of groups, rings and fields, which are essential to further studies in algebra and related topics at Levels 3 and M. The approach will be to study these structures both from an axiomatic basis, but also as a unifying tool for important examples from diverse topics in mathematics. As such, you will need to understand both the abstract theory and its application to a range of examples.

On successful completion of this module, **you should be able to**:

**Module Specific Skills and Knowledge:**

1 recall and apply key definitions in the theory of groups, rings and fields;

2 state, prove and apply core theorems in the theory of groups, rings and fields.

**Discipline Specific Skills and Knowledge**:

3 perform computations accurately;

4 use abstract reasoning to solve a range of problems.

**Personal and Key Transferable / Employment Skills and Knowledge:**

5 communicate your findings effectively in writing;

6 work independently and manage your time and resources effectively.

- review of group axioms and basic examples: cyclic, symmetric and dihedral groups;

- homomorphisms, kernel, image, isomorphisms;

- left and right cosets, normal subgroups;

- quotient groups, the first isomorphism theorem;

- group actions and permutation representations;

- group acting on itself by left multiplication;

- Orbit-Stabiliser Theorem, Orbit Counting Lemma;

- group acting on itself by conjugation, conjugacy classes, centre of a group, conjugacy in Sn, simple groups, A5 is simple;

- Sylow’s theorems;

- axioms for rings, examples: integers, integers modulo n, Matrix ring, polynomial ring (over C, R, and Q);

- units, zero divisors, integral domains, fields, field of fractions of an integral domain;

- rings homomorphisms, kernel, image, characteristic of a ring, p-th power map in characteristic p;

- ideals: principal, prime, and maximal ideals;

- quotient rings, the first isomorphism theorem;

- polynomial rings over a field, and over an integral domain;

- principal ideal domain, unique factorisation domain; maybe: minimal polynomial;

- irreducibility criteria for polynomials: Gauss’s Lemma and Eisenstein’s criterion.

Scheduled Learning & Teaching Activities | 44.00 | Guided Independent Study | 106.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures including example classes |

Scheduled learning and teaching activities | 11 | Tutorials |

Guided independent study | 106 | Lecture and assessment preparation; wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Exercises | One sheet fortnightly | All | Verbal and generic feedback in tutorials |

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours | All | Exam mark – verbal feedback on request |

Coursework – based on questions submitted for formative assessment | 20 | 30 hours in total | All | Annotated script and written feedback |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-assessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

**ELE:** *College to provide hyperlink to appropriate pages*

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Wallace D.A.R. | Groups Rings and Fields | Springer | 2001 | 000-3-540-76177-2 | [Library] | |

Set | Durbin, J. | Modern Algebra: An Introduction | Sixth | John Wiley & Sons | 2009 | 978-0-470-53035-1 | [Library] |

Set | Cameron, P.J. | Fields Introduction to Algebra | Second | Oxford Science Publications | 2008 | 978-0-19-852793-0 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM1706, ECM1701 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Friday 09 January 2015 | LAST REVISION DATE | Friday 09 January 2015 |

KEY WORDS SEARCH | Algebra; rings; fields; groups. |
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