MA204-7-SP-CO:
Abstract Algebra
2019/20
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Postgraduate: Level 7
Current
Monday 13 January 2020
Friday 20 March 2020
15
01 October 2019
Requisites for this module
(none)
(none)
(none)
(none)
MA301
The module introduces the key abstract algebraic objects of groups, rings and fields and develops their fundamental theory. The theory will be illustrated and made concrete through numerous examples in settings that students will already have encountered.
Aims
To introduce basic principles of abstract algebraic structures and to teach how an extensive and important theory can be developed by logical deductions from a small number of axioms.
Learning Outcomes
On completion of the course, students should
• Know and understand the formal definitions for Groups, Rings, and Fields
• be able to produce simple proofs based on the algebraic axioms
• Be familiar with standard examples of these algebras, including the Symmetric Group, Modular Arithmetic, finite abelian groups, Polynomial and Matrix Rings, and examples of finite fields.
• Be familiar with the notions of subalgebras as they apply to Groups, Rings, and Fields
• Understand the notion of isomorphism and homomorphism of these algebra types
• (Level 7) have a comprehensive understanding and appreciation of the theory as above
• (Level 7) be able to produce more difficult arguments in proofs
• (Level 7) understand Euclidean domains, Principal Domains, and Unique Factorization Domains
• (Level 7) ED => PID => UFD theorem
Syllabus
Groups: Binary operations, groups, subgroups, cyclic groups, direct products, groups of permutations, cosets, Lagrange’s theorem; Isomorphisms and homomorphisms of groups
Rings, Fields, zero divisors and integral domains, subrings, ideals. Direct products, homomorphisms, Isomorphisms. The Ring of integers modulo n, polynomial rings, the Euclidean algorithm.
Fields including simple examples of finite fields.
To introduce basic principles of abstract algebraic structures and to teach how an extensive and important theory can be developed by logical deductions from a small number of axioms.
• Know and understand the formal definitions for Groups, Rings, and Fields
• be able to produce proofs based on the algebraic axioms
• Have a comprehensive understanding of standard examples of these algebras, including the Symmetric Group, Modular Arithmetic, finite abelian groups, Polynomial and Matrix Rings, and examples of finite fields.
• Have systematic understanding of the notions of subalgebras as they apply to Groups, Rings, and Fields
• Understand the notion of isomorphism and homomorphism of these algebra types
• Have a systematic understanding of Euclidean domains, Principal Ideal Domains, and Unique Factorization Domains
• Have a critical understanding of the theorem that Euclidean domains are Principal Ideal Domains are Unique Factorization Domains theorem.
No additional information available.
25 lectures and 5 classes (5 lectures, 1 class every fortnight), and 3 revision lectures.
This information can be found on the timetable and module directory.
Level 7: 5 additional lectures on Ring Theory (assessed in final examination).
- Fraleigh, John B. (©2014) A first course in abstract algebra: Pearson.
The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Homework 1 |
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| Coursework |
Homework 2 |
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| Exam |
Main exam: 24hr during Summer (Main Period)
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Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Gerald Williams, email gwill@essex.ac.uk
Professor Gerald Williams (gerald.williams@essex.ac.uk)
Yes
No
Yes
No external examiner information available for this module.
Available via Moodle
Of 234 hours, 65 (27.8%) hours available to students:
169 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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