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.

      If you are registered in tutorial group "sem/online11", you can attend the tutorial through this link as well.  Tutorials will not be recorded.

    • Announcements Forum
    • 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
      413.1KB

      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.

    • Slides shown in lecture File
      269.1KB

      This is one big file of slides for the whole year.

    • 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.

  • Important module information
  • Assessment

  • 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

      Introduction to Algebra has two kinds of assessment:

      • Those not worth marks and provided to support your learning: extra problems, quizzes, etc.
      • Those worth marks: 20% of your total mark for this module will be awarded for coursework, and 80% for the exam. 

      All of these are described below.

      Introduction to Algebra follows the general policies of the School of Maths regarding extenuating circumstances claims and students who get special accommodations from DDS. Refer to the Student Handbook. The module does not use Turnitin.

      • 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 3 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. by asking about situations related to but not identical to the definition in the notes, and what difference the change makes. Overall these parts of 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 2

              • Week 3

              • Week 4

                • Week 4 activities

                  • Submit your solutions to the week 3 coursework by noon on Monday. (The submission form is in the week 3 section.)
                  • Read the sections of the notes featured in this week's lectures:
                    • Monday: Section 2.5.
                    • Tuesday: Section 2.6.
                  • Participate in the lectures:
                    • Monday 2pm: Euclid's algorithm extended. Modular inverses.
                    • Tuesday 4pm: Solving equations in modular arithmetic. Review, questions, examples.
                  • After Tuesday's lecture, try this week's quiz.
                  • Complete the early feedback form. This is optional but it will help me improve my teaching and is a chance for you to reflect on your learning.
                  • Attend your tutorial on Thursday or Friday. Read the tutorial question sheet in advance.

                  Apologies, I forgot to start recording until 30 minutes into Monday's lecture. Help remind me so I don't do that again!

                • Scans from lecture, 14 Feb File
                  1.9MB
                • Scans from lecture, 15 Feb File
                  1.9MB
                • Quiz: modular inverses
                • "Quiz": Extended Euclidean algorithm URL

                  This page will generate an endless supply of questions for you to practice on.

                • Week 4 tutorial questions File
                  70.4KB
                • Week 4 tutorial solutions File
                  85.2KB

              • Week 5

              • Week 6

              • Week 7

                • Week 7 activities

                  • There are no scheduled module activities this week.
                  • If you want something to do: the week 8 coursework is posted, so you could read ahead to Chapter 4 in the notes and start thinking about it. I'll give some tips in the fourth hour of lectures next week.

              • Week 8

              • Week 9

                • Week 9 activities

                  • Submit your solutions to the week 8 coursework by noon on Monday. (The submission form is in the week 8 section.)
                  • This week Dr Fink is away from London. Dr Lewis, one of the tutors, will give the lectures instead. If you have technical difficulties submitting your coursework, please email the Maths office staff, not just Dr Fink as he won't be able to respond quickly. Office hours are cancelled this week. You can still email Dr Fink with questions, bearing in mind the above point about response time.
                  • Read the sections of the notes featured in this week's lectures:
                    • Monday: Section 4.3 and 4.5.
                    • Tuesday: Section 5.1.
                  • Participate in the lectures:
                    • Monday 2pm: Polynomial division. Proof of the polynomial division rule.
                    • Tuesday 4pm: Matrices. Review, questions, examples.
                  • After Tuesday's lecture, try this week's quiz.
                  • Attend your tutorial on Thursday or Friday. Read the tutorial question sheet in advance. 
                • Quiz: polynomial division and roots
                • Solving polynomial equations File
                  91.5KB
                  A 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.
                • A web-toy for exploring complex polynomials and their roots URL
                • Week 9 tutorial questions File
                  81.5KB
                • Week 9 tutorial solutions File
                  105.1KB

              • Week 10

              • Week 11

              • Week 12

              • Exam

                The final exam will be open for 24 hours on QMPlus, but once you start it you will have access to it for three hours. It is designed to take two hours of work to complete.

                The exam will have around ten questions, with mark weighting indicated. Of these there are:

                • Around three questions where you are to write out an answer and upload a PDF, like the coursework. Some of these will be proofs, of standard kinds, closely based on examples from lectures and notes (certainly easier than the coursework).
                • The remainder will be automatically marked questions, like the quizzes in format: multiple choice, numerical answer, fill in the blanks, etc. Some of these will test computational methods. Others will test understanding.
                • How to input things on the exam Quiz

                  On the exam you will have to type in various sorts of objects we have studied on this module. This quiz is here to show you the formats I will use for a few of the more complicated objects. I strongly recommend you try this quiz, so that you don't lose marks on the exam due to a formatting problem.

                  (Because the exam is randomised, you might not actually see a question with every one of these objects.)

                • Semester B final assessment Quiz

                  This quiz assessment is available for a period of 24 hours. Upon accessing the assessment, you will have 3 hours in which to complete and submit it. You may log out and in again during that time, but the countdown timer will not stop. If your attempt is still in progress at the end of your 3 hours (or at the 24-hour deadline if earlier), any answers you have filled will be automatically submitted.

                  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.
                • Interactive mock exam Quiz

                  This is the 2020/21 exam as it would have appeared to students sitting it last year. A few key differences:

                  • The handwritten questions were manually marked. I will not be marking uploads on this mock exam.
                  • You will be shown your marks on the automatically marked questions after you submit. This is not true for the actual exam.