## Topic outline

### 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

### 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, ℤ

. New rings from old: Matrix rings, polynomial rings, complex numbers._{m}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.

### Week 2

### Week 3

### Week 4

### Week 5

### Week 6

### Week 6 activities

- Submit your solutions to the week 5 coursework by noon on Monday. (The submission form is in the week 5 section.)
- Read the sections of the notes featured in this week's lectures:
- Monday: Section 3.4.
- Tuesday: Section 3.5.
- Participate in the lectures:
- Monday 2pm: Rings from modular arithmetic. Consequences of the ring axioms.
- Tuesday 4pm: More consequences of the ring axioms. 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.

### 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.

- There are no scheduled module activities this week.

### Week 8

### Week 8 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 4.1 and 4.2.
- Tuesday: Section 4.4.
- Participate in the lectures:
- Monday 2pm: Polynomial rings. Properties of polynomial rings.
- Tuesday 4pm: Roots and factors. 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.
- Complete the coursework for this week.

### 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.

### Week 10

### Week 10 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 5.2 and 6.1.
- Tuesday: Section 6.2 and 6.3.
- Participate in the lectures:
- Monday 2pm: Matrix rings. Permutations.
- Tuesday 4pm: Cycles. 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.

- Complete the coursework for this week.

### Week 11

### Week 11 activities

- Submit your solutions to the week 10 coursework by noon on Monday. (The submission form is in the week 10 section.)
- Read the sections of the notes featured in this week's lectures:
- Monday: Section 7.1.
- Tuesday: Section 7.2.
- Participate in the lectures:
- Monday 2pm: Groups. First examples of groups.
- Tuesday 4pm: Cayley tables. 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.

### Week 12

### Week 12 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 7.3 and 7.4.
- Tuesday: Section 7.5.
- Participate in the lectures:
- Monday 2pm: Consequences of the group axioms. The group of units.

- Tuesday 4pm: Subgroups. About the exam. Review, questions, examples.
- There is no standard tutorial this week. Instead there will be a revision lecture based on past exam papers, to be led by Dr Matthew Lewis. This will take place
*Monday from 6pm to 7:30pm*. It is online only; you can find it in the usual online lecture room.

- After Tuesday's lecture, try this week's quiz.
- Complete and submit the coursework for this week. Because of Easter, the due date for this coursework is noon on
*Tuesday 19 April*, next week, after the end of the teaching term.

Because there will be no tutorial in week 12, I have not prepared a tutorial question sheet. There are a couple questions on groups in the week 11 tutorial questions; do those if you haven't yet. For more, see the extra questions for chapter 7. (Remember that there are extra questions sheets for every chapter under the "Assessment" tab.)

### Exam