Topic outline
- Online lecture room External tool
Introduction to Algebra is being taught in the "mixed mode education" paradigm. You can attend each lecture for this module, as you prefer,
- on campus, in the Mason lecture theatre; or
- remotely, using Blackboard Collaborate, using the link above.
The above link also lets you view recordings of past lectures.
- Student forum
This forum is available for everyone to post messages to. If you have any questions about the module that are not private, please ask them here. I will be monitoring it to answer your questions. It is OK if you email me questions too, but by using the forum other students can benefit from the answer.
You can add maths notation in your posts using the button with a calculator icon.
I have left last year's forum messages in place, just in case they are useful to read through.
- Lecture notes File
Full lecture notes for the module are given here, and will serve as a definitive record of what is examinable.
I may edit these notes as the semester progresses but only in minor ways, e.g. to correct typos or improve explanations. I will not add new content, and the topics and numbering of sections in the notes will not change.
Information about what is taught in each week can be seen in the "Module Content" tab below, in the sections for the individual weeks. After lectures I will scan the papers I wrote on during the lecture and upload those under the header for the week as well. Some weeks also have an appendix on a topic going beyond the notes, labelled as "not examinable", in case you're interested.
My intention is that you will not have to take your own notes during the lectures. But some students find it helps their learning to take notes anyway, or to download or print my notes and add their own annotations during the lecture. Think about whether this is you.
What is examinable?
If I skip some material in the lecture notes, I will annotate it as not examinable in the PDF, and add it to this section of the page. Everything else in the lecture notes is examinable.
Module activities
Each weekly section on the QMPlus page starts with a list of the activities which I expect you to undertake that week.
Lectures
Introduction to Algebra will be taught in "mixed mode education" this year. Real-time lectures will take place twice a week. If you are joining remotely, this will be done via Blackboard Collaborate, and you can ask questions via the chat.
In the first three hours each week I will teach new content. The fourth hour will be dedicated to questions, clarifications, examples, model solutions, etc. of the content from the preceding week, and miscellaneous other topics such as exam guidance. Please read over the week's notes before the Tuesday lecture so that you can come prepared with questions on things you don't understand.
Tutorials
You will have a timetabled tutorial class each week, except weeks 1 and 7. This will be in a small group of roughly 15 students. The function of the tutorial is to give you a place to work on questions in a setting where a tutor is present to help you. Each tutorial will have a dedicated problem sheet. You may work in groups in your tutorials.
The tutor who runs your tutorial will also mark your coursework. Your tutor will leave written comments on your work, and if you have questions about these comments or how your coursework was marked, you should ask your tutor in the first instance.
That is, you must:
- Attend your timetabled tutorial and participate actively in it by working on the tutorial questions.
- Submit the assigned coursework questions for feedback.
It's OK if you need help with the tutorial questions. But you mustn't give up on a question because you don't see how to solve the whole thing right away. Try various approaches to the question, and if none of them succeed, then at least you'll be in a position to have an informative conversation with your tutor about what to do next.
Assessment
There will be five assessed courseworks, each counting for 4% of your module mark. These will appear at fortnightly intervals, skipping week 7, and you will have at least a week to complete each one. The final exam in May will count for 80% of your module mark.
See the Assessment tab for details.
Assessment Information
QMplus Quizzes
The QMPlus quizzes linked in the sections for each week's content provide you immediate feedback on your answers and are meant for consolidation and/or revision after you have worked through each week's lectures and notes. They do not count towards your module mark and are provided to support your learning.
For a few computational topics for which it's harder to get a QMPlus quiz to generate random questions, I have also included links to simple webpages to generate more.
Extra questions
Coursework Assignments
There will be five coursework assignments, each worth 4% of your module mark. They belong to Weeks 3, 5, 8, 10, and 12 of the semester, but will be due at 12:00 noon on Monday the following week. That is, the due dates are
- Monday 14 February
- Monday 28 February
- Monday 21 March
- Monday 4 April
- Monday 18 April
Your solutions are to be uploaded as a single pdf to the QMPlus assignment in the week in question. Your tutor will mark them and provide feedback in time for your tutorial. I have chosen the deadline so that it doesn't clash with the Calculus II deadlines or any first-year Maths lectures, and so that your tutors have enough time for the marking.
Final exam
Format of the final exam
The format of the final exam will be a QMPlus quiz, like those you sat in the January exam period. The exam will be available for 24 hours, but you will have 2 hours from the time you first open the exam to complete it.
In an ordinary year, the exam for this module has several questions like "write down the definition of X" or "write down the proof of Theorem Y". Because you will have access to your notes during the exam, it will not feature straight bookwork questions like these. I may examine definitions and theorems in other ways, e.g. "which of the above is equivalent to X?" "which of the above would contradict Y?" Or I may give you two re-worded definitions, but one of them will be inaccurate, and I'll ask you to say which one, and why. Overall the exam will require more thought and the ability to use your knowledge.
Past papers
The exam in 2020 and 2021 were QMPlus quizzes. In 2020 the course of lectures was interrupted by industrial action so not all topics were examinable.
- Past exam papers Folder
In 2019 and earlier, Introduction to Algebra was examined the traditional way, on paper, and the exam papers are collected here. In 2020 and 2021, the exam was a randomised QMPlus quiz, and in this folder I have put one random variation each for those years.
Past papers may be for earlier versions of the module syllabus, and as such their topic coverage and/or rubric might not match this year's exam. In particular, the following are not examinable:
- Extracting roots of complex numbers, i.e. solving zn = a in the complex numbers.
- Pseudocomplex numbers, hypercomplex numbers, quaternions.
- Isomorphism.
- Cosets.
- Fermat's little theorem.
- Mathematical induction "for its own sake", like Q1 on the 2017 paper. You are expected to know mathematical induction; it is used in some of our proofs. But earlier versions of this module spent a few lectures at the beginning just revising
induction, which is no longer done.
Syllabus
1. The integers: revision of divisibility, gcd, Euclid's algorithm, and primes. Prime factorisation.
2. Polynomials: Real polynomials, divisibility, factorisation, irreducible polynomials, factorisation, roots over ℝ and ℂ, statement of Fundamental Theorem of Algebra.
3. Equivalence relations and congruence: Revision of relations. Equivalence relations and partitions. Congruence modulo m, modular arithmetic.
4. Rings and fields: Binary operations. Definitions and basic properties of rings and fields. Examples: ℤ, ℚ, ℝ, ℂ, ℤm. New rings from old: Matrix rings, polynomial rings, complex numbers.
5. Permutations: identity, associativity, inverses, cycle decomposition
6. Groups: Groups and subgroups; examples including the symmetric group, symmetry groups. Statement of Lagrange's theorem.
Learning outcomes
Academic Content
This module covers:
- Polynomials and their factorization.
- Equivalence relations and partitions. Modular arithmetic.
- An introduction to rings and fields.
- An introduction to groups including statement of Lagrange’s Theorem.
- Permutations and the symmetric group.
Disciplinary Skills
At the end of this module, students should be able to:
- Recognise and construct a valid proof, and use counterexamples to disprove assertions.
- Understand the difference between necessary and sufficient conditions.
- Understand and use the relation between equivalence relations and partitions.
- Perform the division and Euclidean algorithms on integers and polynomials.
- Know the definitions of group, ring and field, and deduce some consequences of the axioms for these structures.
Attributes
At the end of this module, students should have developed with respect to the following attributes:
- Grasp the principles and practices of their field of study.
- Acquire substantial bodies of new knowledge.
- Explain and argue clearly and concisely.
- Acquire and apply knowledge in a rigorous way.
- Connect information and ideas within their field of study.
Week 1
Week 1 activities
- Read the "Important module information" and "Assessment" tabs on this QMPlus page.
- Read the advice on how to study for this module, and make a plan for your own study throughout the semester.
- Read the sections of the notes featured in this week's lectures:
- Monday: Section 0.1 and 1.1.
- Tuesday: Section 1.2.
- Participate in the lectures:
- Monday: Introduction to the module. What is algebra? Ordered pairs and Cartesian product.
- Tuesday: Relations as subsets of the Cartesian product. Review, questions, examples.
- After Tuesday's lecture, try this week's quiz.
- Read the "Important module information" and "Assessment" tabs on this QMPlus page.
- On the meaning of "natural numbers" File
I briefly discuss the two different definitions that different mathematicians give to ℕ. Not examinable.
- Functions as relations File
I define functions formally as a kind of relation. Not examinable.
Week 2
Week 2 activities
- Read the sections of the notes featured in this week's lectures:
- Monday: Section 1.3.
- Tuesday: Section 2.1.
- Participate in the lectures:
- Monday: Partitions. Translating between relations and partitions.
- Tuesday: Integer division. Review, questions, examples.
- After Tuesday's lecture, try this week's quiz.
- Tutorials start this week. Attend yours on Thursday or Friday (check your timetable).
- Week 2 tutorial solutions File
Week 3
A few YouTube videos on how to write proofs:
- Trefor Bazzet:
- The Math Sorcerer:
- Trefor Bazzet:
Week 4
- "Quiz": Extended Euclidean algorithm URL
This page will generate an endless supply of questions for you to practice on.
Week 5
- The quaternions File
The quaternions are a skewfield that extends the field of complex numbers. Not examinable.
Week 6
Week 7 (Reading Week)
Week 8
- Solving polynomial equations FileA discussion of how, and whether, you can compute the roots of a polynomial f ∈ K[x] of degree 2 or greater where K is a field (and in particular K = ℂ). Not examinable.
Week 9
Week 10
Week 11
- "Quiz": permutation computations URL
The quiz facility on QMPlus can't deal well with permutations. So this is an endless supply of questions on computing with permutations.
Week 12
- Cosets and Lagrange's theorem File
A proof of Theorem 7.10 from the lecture notes. Not examinable.
- Odd and even permutations and the alternating group File
I introduce the alternating group, a subgroup of symmetric group consisting of all the "even" permutations. Not examinable.
Revision lecture
There will be a one-hour revision lecture at 12:00 noon on Tuesday 4 May, in the online lecture room. Please use the poll below to tell me which topics you would like me to discuss.
Exam
The final exam will be open for 24 hours on QMPlus (like your semester A exams). It is designed to take about three hours of work to complete.
The exam will have ten questions, worth ten marks each. Of these there are:
- Two questions where you are to write out an answer and upload a PDF, like the coursework. These will be examples or proofs of standard kinds, closely based on examples from lectures and notes (certainly easier than the coursework). I will mark these.
- Eight automatically marked questions, like the quizzes in format: multiple choice, numerical answer, fill in the blanks, etc. Some of these are based on computational methods. Others will, to steal Dr Johnson's turn of phrase, test understanding rather than memorisation. If there are any complicated types of answers you need to type in, I will provide a little sample quiz so you can be sure you're typing them in in the correct format.
- How to input permutations Quiz
There is a question on the exam for which you must enter a permutation. This quiz is to demonstrate the format in which you will have to type the permutation in, so that you won't be caught out on the exam by little issues of formatting.
- Late-summer reassessment 2020/21 Quiz
This exam has 10 questions, all of which are on Page 1 of the quiz. Page 2 is for uploading your rough work.
This quiz assessment is available for a period of 24 hours, and you may log out and in again during that time. If your attempt is still in progress at the 24-hour deadline, the answers you have filled in so far will be automatically submitted. You are also invited to upload scans of your rough work; in the event that you fail the assessment, we will use these to see whether there are method marks we can award to help you pass.
In completing this assessment:
- You may use books and notes.
- You may use calculators and computers, but you must show your working for any calculations you do.
- You may use the Internet as a resource, but not to ask for the solution to an exam question or to copy any solution you find.
- You must not seek or obtain help from anyone else.
- Late-summer reassessment 2020/21: version for 2019/20 candidates Quiz
This exam has 10 questions, all of which are on Page 1 of the quiz. Page 2 is for uploading your rough work.
This quiz assessment is available for a period of 24 hours, and you may log out and in again during that time. If your attempt is still in progress at the 24-hour deadline, the answers you have filled in so far will be automatically submitted. You are also invited to upload scans of your rough work; in the event that you fail the assessment, we will use these to see whether there are method marks we can award to help you pass.
In completing this assessment:
- You may use books and notes.
- You may use calculators and computers, but you must show your working for any calculations you do.
- You may use the Internet as a resource, but not to ask for the solution to an exam question or to copy any solution you find.
- You must not seek or obtain help from anyone else.
- Semester B final assessment 2020/21 Quiz
This quiz assessment is available for a period of 24 hours, and you may log out and in again during that time. If your attempt is still in progress at the 24-hour deadline, the answers you have filled in so far will be automatically submitted. You are also invited to upload scans of your rough work; in the event that you fail the assessment, we will use these to see whether there are method marks we can award to help you pass.
In completing this assessment:
- You may use books and notes.
- You may use calculators and computers, but you must show your working for any calculations you do.
- You may use the Internet as a resource, but not to ask for the solution to an exam question or to copy any solution you find.
- You must not seek or obtain help from anyone else.
- Semester B final assessment 2020/21: version for 2019/20 candidates Quiz
This quiz assessment is available for a period of 24 hours, and you may log out and in again during that time. If your attempt is still in progress at the 24-hour deadline, the answers you have filled in so far will be automatically submitted. You are also invited to upload scans of your rough work; in the event that you fail the assessment, we will use these to see whether there are method marks we can award to help you pass.
In completing this assessment:
- You may use books and notes.
- You may use calculators and computers, but you must show your working for any calculations you do.
- You may use the Internet as a resource, but not to ask for the solution to an exam question or to copy any solution you find.
- You must not seek or obtain help from anyone else.