Topic outline

    • Online lecture room External tool
    • Student forum
    • Lecture notes File
    • Slides shown in lecture File
  • Module activities

    Each weekly section on the QMPlus page starts with a list of the activities which I expect you to undertake that week.


    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.


    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.


    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
          • 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.


              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

                • "Quiz": Extended Euclidean algorithm URL
              • Week 5

              • Week 6

              • Week 7

              • Week 8

              • Week 9

                • Solving polynomial equations File
              • 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
                • Semester B final assessment Quiz
                • Interactive mock exam Quiz