## Topic outline

**IMPORTANT CORRECTION**to the introductory video: The lectures will take place on MS Teams, not on Blackboard Collaborate as announced in the video. (As a backup I'll use Zoom, details to be announced if it becomes necessary.)

### Online learning

Because of the ongoing Covid-19 pandemic, this module will be taught primarily online this year. Please watch the introductory video to find out more about how things will work and how to get the most out of the module. Here are some key points.

Lectures

Real-time lectures will take place four times a week via Blackboard Collaborate. A link for each lecture appears on the QMplus page, and you can join to see on-screen lectures. You can ask questions via the chat or verbally.

In the first three lectures each week, Monday to Wednesday, I will teach new content. The fourth lecture will be dedicated to review, further clarifications, examples and model solutions of the content from the preceding week, and miscellaneous other topics such as exam guidance. Please read over the week's notes after the Wednesday lecture so that you can come prepared with questions.

Tutorials

You will have a timetabled tutorial class in weeks 3, 5, 8, 10, and 12. This will be in a small group of roughly 12 students. There will be a written assignment associated with each tutorial. One question on each assignment will be labelled as to be submitted for marking by your tutor. You should also attempt the other questions before the tutorial. In the tutorial, your tutor will answer any questions you have on the material and give individual feedback on your work.

You must:

- Attend your timetabled tutorial and participate actively in it.
- Submit the assignment question for feedback before each tutorial.
- Make dedicated attempts at the other assignment questions and have your written answers with you in the tutorial.

It's OK if you need help with the assignment 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 5% of your module mark. These will appear at roughly two-week intervals, and you will have a week to complete each one. The final exam in January will count for 75% of your module mark.

See the Assessment tab for more information.

Here is a link to the introductory video.

### 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: 25% of your total mark for this module will be awarded for coursework, and 75% 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

The first lot of files below are big lists of questions you can try solving if you want more practice questions than are available on the exercise sheets. I have sorted them by chapter and will be releasing more chapters as the semester continues.

### Coursework Assignments

There will be five coursework assignments, each worth 5% of your module mark. They will be due at 5:00pm on Tuesday in Weeks 3, 5, 8, 10, and 12 of the semester. Your solutions are to be uploaded as a single pdf to the QMPlus assignment. Your tutor will mark them and provide feedback in time for your tutorial on Thursday or Friday of the same week.

### Final exam

Format of the final exam

The exact format of the final exam is not yet decided; it depends on how Covid-19 restrictions evolve. However, the most likely format is a QMPlus quiz, similar to those used for first-year modules in the January exam period.

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. Instead I will examine definitions and theorems in other ways, e.g. "which of the above is equivalent to X?" "which of the above would contradict Y?" Or I may give you two re-worded definitions, but one of them will be inaccurate, and I'll ask you to say which one, and why. Overall the exam will require more thought and the ability to use your knowledge.

Past papers

The 2020 exam was a QMPlus quiz, but the course of lectures was interrupted by industrial action so not all topics were examinable.

### 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

- Watch the introductory video.
- 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.
- Tuesday: Section 1.1.
- Wednesday: Section 1.2.
- Participate in the live lectures, in the online lecture room:
- Monday 11:30am: What is algebra?
- Tuesday 4pm: Ordered pairs and Cartesian product.
- Wednesday 10am: Relations as subsets of the Cartesian product.
- Friday noon: Review, questions, examples.
- After Wednesday's lecture, try this week's quiz.

### Week 2

### Week 2 activities

- Read the sections of the notes featured in this week's lectures:
- Monday–Tuesday: Section 1.3.
- Wednesday: Section 2.1.
- Participate in the live lectures, in the online lecture room:
- Monday 11:30am: Partitions.
- Tuesday 4pm: Translating between relations and partitions.
- Wednesday 10am: Integer division.
- Friday noon: Review, questions, examples.
- After Wednesday's lecture, try this week's quiz.
- The submission question on the Week 3 coursework, below, is on Week 2 material, so I suggest starting to think about it this week.

### Week 3

### Week 3 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 2.2.
- Tuesday: Section 2.3.
- Wednesday: Section 2.4.
- Participate in the live lectures, in the online lecture room:
- Monday 11:30am: Congruence mod m.
- Tuesday 4pm: Arithmetic with congruence classes.
- Wednesday 10am: GCD and Euclid's algorithm.
- Friday noon: Review, questions, examples.
- After Wednesday's lecture, try this week's quiz.
- Submit your coursework by 5pm Tuesday.
- If you need a refresher on how to write proofs for the coursework, I recommend "How to write proofs" by Eugenia Cheng.
- Attempt the other questions on the coursework sheet before your tutorial.

### Week 4

### Week 4 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 2.5.
- Tuesday / Wednesday: Section 2.6.
- Participate in the live lectures, in the online lecture room:
- Monday 11:30am: Euclid's algorithm extended.
- Tuesday 4pm: Modular inverses.
- Wednesday 10am: Solving equations in modular arithmetic.
- Friday noon: Review, questions, examples.
- After Wednesday's lecture, try this week's quiz.
- Part (a) of the submission question on the Week 5 coursework, below, is on modular arithmetic. I suggest starting to think about it this week. (For (a)(i) you can at least work out the proof now.)

### Week 5

### Week 5 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 3.1.
- Tuesday: Section 3.2.
- Wednesday: Section 3.3.
- Participate in the live lectures, in the online lecture room:
- Monday 11:30am: The axiomatic method. Rings and fields.
- Tuesday 4pm: Understanding the axioms.
- Wednesday 10am: The complex numbers.
- Friday noon: Review, questions, examples.
- After Wednesday's lecture, try this week's quiz.
- Submit your coursework by 5pm Tuesday.
- Attempt the other questions on the coursework sheet before your tutorial.

### Week 6

### Week 7 (Reading Week)

### Week 8

### Week 9

### Week 10

### Week 11

### Week 12