Algebraic structures I
|No revision lecture has been scheduled for this course during the revision "week".
If you want help with your revision, you are welcome to visit me during my office hours, or email for
I believe the exam is from 10-12 on Thursday 2nd June, in York Hall (for most people), but YOU SHOULD
CHECK YOUR PERSONAL TIMETABLE.
I am unlikely to be available for last-minute questions after Tuesday 24th May, as I have another exam to mark.
26/04/11: Selected solutions to Exercises 10 are now available from the
29/03/11: Solutions to Exercises 9 are now available from the
I have now finished marking the mid-term test. Marked scripts, and model
solutions, will be available in the tutorial today.
18/03/11: The slides for lecture 25 (Monday 14/3/11) can be found
from the link below.
11/03/11: Exercises for week 9: details below. Hand in Q.50.
11/03/11: An example of a PID that is not a Euclidean domain:
sketch of proof.
08/03/11: Solutions to Exercises 5 and 6 are now available below.
21/01/11: At your request, the deadline for handing in coursework
has been extended to Monday at 4pm.
20/01/11: Summary notes of the first five lectures are available from the
20/01/11: Friday tutorial now confirmed as 11am, in Eng 324.
17/01/11: Exercises for this week are listed below.
13/01/11: Subject to being able to find a suitable room, the
tutorial scheduled for 9am on Friday will be moved to 11am on Friday, to avoid
the clash with Number Theory.
13/01/11: The lecture at 4pm on Thursday will from now on be in the
new Mathematics Lecture Theatre (on the right immediately as you enter the basement
entrance of the mathematics building).
11/01/11: Exercises for the whole term are available from the link below,
as are summary notes for the first lecture.
10/09/10: Course information in preparation.
In the meantime, Prof. Cameron's Notes for MAS201 from 2006
provide the best guide as to
what is likely to be in the course.
- Value: 15 credits
- Level: 5
- Semester: B
- Prerequisites: MTH4104 Introduction to algebra, essential.
Course descriptions and syllabus
- Lectures: Tu10 in Queen's EB1; Th4 in Geog 226 (note another change!); F10 in Queen's EB1.
- Lecturer: Professor Robert Wilson, Room G51. For office hours, see my homepage.
- Exercise classes: either Th2 in Queen's EB1, or F11 (note change!) in Eng 324, starting in week 2.
- Solutions to exercises to be handed in to the BLUE box on the GROUND floor of the MATHEMATICS building by MONDAY 4pm.
Your working week
Each module is designed to fill one-quarter of a full-time working week.
That is, you are expected to work roughly ten hours a week on each module.
Only four hours are timetabled. The rest is your responsibility. It should include,
at the very least:
reading your lecture notes and/or the textbook,
- doing exercises,
preferably on your own, and
- going over the feedback
provided on your attempts, to see where you can improve.
Lectures, notes, and books
What is the purpose of lectures, if the notes are available online? Experience has shown
that it is actually quite difficult to learn mathematics by reading notes or books,
even if you are conscientious about doing the exercises (which most people
If you are studying music, it is much better to hear it in real time, rather
than read the score. A mathematical proof is like a piece of music:
there is really no substitute for seeing it develop in real time.
If I have time, I may put up some summary lecture notes here from time to time.
A summary of Lectures 1-29 is currently available.
For the rest of the course, or for an alternative view, you are recommended to
Prof. Cameron's Notes for MAS201 (the previous version of MTH5100) from 2006
An example of a PID that is not a Euclidean domain:
sketch of proof.
Lecture 25 slides.
Exercises, tutorials, and feedback
Exercises and tutorials
These should be regarded as compulsory. Mathematics is not about
learning facts ("know-what"), it is about learning methods ("know-how").
Methods and techniques cannot be learnt without practice.
As Confucius (551-479BCE) said:
"I hear and I forget;
Mathematics is about doing, not about hearing or seeing.
I see and I remember;
I do and I understand."
Here are some Exercises,
which are intended to last the whole semester. Not all of these questions will be set
formally, but they are provided for your benefit anyway. Each week's set of exercises
is structured so that there are:
(a) practice questions, which you can get help on
(b) one feedback question, to be handed in and marked for feedback, for
which help is not generally available in tutorials, since this defeats the purpose of feedback,
(c) extra questions for those who want to deepen their learning.
Marked work will normally be returned for feedback in the tutorials 3/4 days after handing in.
- Exercises for week 2 tutorials:
practice questions 1, 2, 7; feedback question 9, for handing in by Monday 12 noon in Week 3; extra questions for keen students 8, 10
(and any others in 1-12). The learning objectives for this week are to consolidate
the pre-requisite material on sets, functions, operations, and relations.
- Exercises for week 3 tutorials:
practice questions 13-16,19; feedback question 20, for handing in by Monday 4pm in Week 4; extra questions for keen students:
any others in 13-24. The learning objectives for this week are to learn the definitions of various
types of ring (commutative,
with identity, division ring, etc.); to be able to deduce standard properties
like 0.x=0; to be able to apply the definitions to determine whether certain
objects are rings; and to become familiar with a number of important
examples of rings.
- Exercises for week 4 tutorials:
questions 23-27,29-30; feedback questions 25(a),(c),(e),(g), and the same parts of Q29,
for handing in by Monday 4pm in Week 5. The learning objectives for this week
are to learn and understand the formal definitions of
matrix rings and polynomial rings; to learn the definition of subring and be able
to apply the subring tests to subsets of rings; and to understand the concept of
a coset of a subring.
- Exercises for week 5 tutorials:
practice questions 32,33,34(a),37; feedback question 36(a,b,c,d), but with justification,
for handing in by Monday 4pm in Week 6. The learning objectives for this week are
to learn the definitions of homomorphism and isomorphism,
image and kernel, and to be able to recognise homomorphisms and compute their
kernels and images; to be able to tell whether or not a given subset of a ring is an ideal,
and to construct and work in the corresponding quotient ring.
- Exercises for week 6 tutorials: 37 (assuming you did not do it last week),
38, 39, 40, 41(a,b,c). In Q.40, an ideal I in a ring R is called maximal if the only ideals containing I are I and R;
In (d) and (e), J is used for the ring of Gaussian integers, that is complex numbers a+bi where a and
b are ordinary integers.
Hand in your solution to Q.41 by Monday 4pm in Week 8.
- Exercises for week 8 tutorials: 43,44,47,48,52. In Q.43(c) it should say 'Show that S...',
not 'Show that R...'. In Q.43(b) the notation may not be quite clear: it means that if
X is not equal to the empty set or to U, then X is a zero-divisor.
Hand in your solution to Q.52 by Monday 4pm in Week 9.
- Exercises for week 9 tutorials: 46,49,50,53,56. In several of these questions, J denotes the
ring of Gaussian integers.
Hand in your solution to Q.50 by Monday 4pm in Week 10.
- Exercises for week 10 tutorials: 57,58,59,60,63. In several of these questions, J denotes the
ring of Gaussian integers. There is a misprint in 58(b): F should be replaced by R.
Hand in your solution to Q.58(b) by Monday 4pm in Week 11.
- Exercises for week 11 tutorials: 64,66,67,68(b)-(e).
Hand in your solution to Q.68 by Monday 4pm in Week 12. Selected solutions.
- Exercises for week 12 tutorials: 70,71,73,74,77,83. Although there is not time during the
term to hand in any of these for feedback, it is important that you attempt them
in order to learn the material on groups in the course, which will be tested
in the exam. Tutorials will be held as usual this week to provide help for you. Selected solutions.
Note: this is a change from last year
20% of the assessment for this course is based on the mid-term test.
80% of the assessment is on the final exam in May/June.
Note: this does not mean that the weekly exercises are optional!
They are still compulsory, and anyone not making a genuine attempt at them
may be excluded from the course, including from the exam.
- Exams from 2007
Note however that, due to changes in the School regulations,
this year's exam is in a different format. Other past exams can be found on the library web-site.
- These and other past exams can be found on the library web-site.
- Sample exam here.
- Last year's mid-term test is here
I will not provide model solutions to past examination papers,
because they encourage poor quality learning. If you wish to use
past exam papers as an aid to your revision, you may bring your
solutions to one of my office hours, and I will
give you feedback on them as far as time allows.
Other course material
Some thoughts on the overall aims of the course,
my teaching strategy, and study skills,
are collected here.
The recommended course text is
P. J. Cameron, Introduction to Algebra. Oxford University Press.
- Prof. Cameron's Notes for MAS201 (the previous version of MTH5100) from 2006
Frequently asked questions
- Can we have more examples?
The module consists almost entirely of examples as it is.
- Can we have model solutions to last year's exam?
No. This only encourages a bad approach to learning.
- Will there be proofs in the exam?
Yes. Proof is what mathematics is all about.
- How many of the coursework marks count towards the final mark?
None. The `continuous assessment' part of the assessment is based entirely
on the mid-term test.
- What is the format of the exam?
In accordance with new guidelines, there will be no choice on the exam.
The precise number and length of questions may vary, but there are likely to
be six questions, distributed among the major topics covered in the course
roughly according to the number of lectures.
Robert A. Wilson
Created 10 August 2009
Updated 26 April 2011