## 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

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 2 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. "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 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.

### 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 2 activities

- Read the sections of the notes featured in this week's lectures:
- Monday: Section 1.3.
- Tuesday: Section 2.1.
- Participate in the lectures:
- Monday: Partitions. Translating between relations and partitions.
- Tuesday: Integer division. Review, questions, examples.
- After Tuesday's lecture, try this week's quiz.
- Tutorials start this week. Attend yours on Thursday or Friday (check your timetable).

### Week 3

A few YouTube videos on how to write proofs:

- Trefor Bazzet:

- The Math Sorcerer:

- Trefor Bazzet:

### Week 4

### Week 5

### Week 6

### Week 7 (Reading Week)

### Week 8

### Week 9

### Week 10

### Week 11

### Week 12

### Revision lecture

There will be a one-hour revision lecture at

**12:00 noon**on**Tuesday 4 May**, in the online lecture room. Please use the poll below to tell me which topics you would like me to discuss.### Exam

The final exam will be open for 24 hours on QMPlus (like your semester A exams). It is designed to take about three hours of work to complete.

The exam will have ten questions, worth ten marks each. Of these there are:

- Two questions where you are to write out an answer and upload a PDF, like the coursework. These will be examples or proofs of standard kinds, closely based on examples from lectures and notes (certainly easier than the coursework). I will mark these.
- Eight automatically marked questions, like the quizzes in format: multiple choice, numerical answer, fill in the blanks, etc. Some of these are based on computational methods. Others will, to steal Dr Johnson's turn of phrase, test understanding rather than memorisation. If there are any complicated types of answers you need to type in, I will provide a little sample quiz so you can be sure you're typing them in in the correct format.