Topic outline

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  • 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

    • Extra questions

    • Coursework Assignments

    • Final exam

    • 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 5

        • Week 6

        • Week 7

        • Week 8

        • Week 9

          • Solving polynomial equations File
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        • Week 10

        • Week 11

        • Week 12

        • Exam