Unit information: Algebra 2 in 2015/16

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Unit name Algebra 2 MATH21800 20 I/5 Teaching Block 2 (weeks 13 - 24) Professor. Tim Dokchitser Not open MATH11511 Number Theory & Group Theory and MATH11005 Linear Algebra and Geometry (From 2015/16: Foundations and Proof, Introduction to Group Theory, Linear Algebra and Geometry) None School of Mathematics Faculty of Science

Description

Unit aims

To develop the theory of commutative rings, and to apply it to solving problems concerning the factorisation of polynomials, algebraic numbers, ruler-and-compass constructions, and the construction of roots of polynomials.

General Description of the Unit

Algebraic structures -- such as groups, rings, and fields -- are prevasive in mathematics. This course focuses on (commutative) rings, which are sets equipped with two (commutative) operations (called addition and multiplication), and that contain an additive identity and an additive inverse for each element of the set. A fundamental example of a ring is Z, the set of integers; other important examples include Q, Z modulo n, and Q[X], which is the set of polynomials in X with rational coefficients. A fruitful way to study rings and their properties is to study "homomorphisms" between rings: a homomorphism is a map that preserves addition and multiplication (just as a linear transformation preserves vector addition and scalar multiplication). Using homomorphisms and generalised modular arithmetic, we develop means of determining when a ring has additional nice properties, such as having multiplicitive inverses for each nonzero element of the ring. This is a very beautiful and clean theory; in proving the theorems, the students will learn some new techniques and strengthen their proof-writing skills.

Relation to Other Units

This unit has some relationship to (but is independent of), Linear Algebra 2 and the Level 7 unit on Representation Theory, and has a stronger relationship to Algebraic Number Theory (Level 6) and Galois Theory (Level 7). It provides the foundation for Galois Theory.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Intended learning outcomes

Learning Objectives

After taking this unit, students should be able to state the basic definitions and results in the subject, to utilise the fundamental proof techniques, and to solve problems similar to those worked in the lectures and set as homework.

Transferable Skills

The ability to understand and apply general theory, and the acquisition of facility in calculating in a variety of number-systems.

Teaching details

There are 3 lecture classes and 1 problems/feedback session each week. The course is based on the lectures and exercises. The basic lecture notes will be posted and solutions to the assigned exercises will be distributed.

Solutions to most exercises not assigned will be presented in lectures, but typically these solutions will not be posted or distributed.

There are 5 in-class 10 minute quizzes (open-book, open-note). The solutions to the quizzes are discussed in the problems classes, as are common difficulties with the assigned exercises. The quizzes and homework are designed to help the students gain aptitude and confidence with the material and techniques as the term progresses.

Solutions to previous exams will not be posted or distributed.

The last 2 weeks of the course are devoted to review and revision, and in this time exercises (both assigned and not assigned) and previous exam questions will be addressed.

Besides the problems classes, there is also a weekly office hour during which students can ask questions about lectures and exercises.

Assessment Details

85% Examination, 10% Homework Questions, and 5% Open-book Quizzes.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.
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