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

    • 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 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