## Algebraic Structures 2

 Unit code: MATH20212 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

MATH20212 pre-requisites

#### Aims

The course unit unit aims to introduce the algebraic structures of rings and fields; describe the quotient structure and its connection with homomorphisms of rings; present important examples rings and develop some of their properties with particular emphasis on polynomial rings and factorisation in rings.

#### Overview

This course builds on Algebraic Structures 1, which is a prerequisite, and continues the strong emphasis on examples.

The algebraic structures of rings and fields will be introduced. The construction of quotient rings and the relationship with homomorphisms is one of the main themes. These ideas will be used to construct roots of polynomials in extension fields. Factorisation in polynomial rings and rings of integers of number fields will also be studied.

#### Learning outcomes

On completion of this unit successful students will be able to:

• define rings, domains and division rings, and describe standard examples,
• state and prove properties of rings and apply these to standard examples,
• describe and recognise special types of elements in rings including zero divisors, units, nilpotents, idempotents and irreducible elements,
• define and recognise homomorphisms of rings and state and prove properties of homomorphisms,
• define an ideal of a ring and state and prove properties of ideals,
• define a factor ring, state and prove properties of factor rings and construct factor rings,
• describe properties of polynomial rings, calculate greatest common divisors and factorize polynomials in K[X] where K is a field,
• state and prove Kronecker's Theorem and use this to construct extension fields.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework; Weighting within unit 20%
• 2 hours end of semester examination; Weighting within unit 80%

#### Syllabus

1.Definitions and examples of rings (rings of numbers, rings of matrices, quaternions, rings of endomorphisms, group rings, rings of polynomials, subrings); [4 lectures]

2.Domains, fields and division rings; nilpotent and idempotent elements, products of rings; (many) examples; with students gaining familiarity with the ideas and examples through attempting exercises. [4]

3.Isomorphisms and homomorphisms (of rings): what is preserved and reflected; kernel of a homomorphism, ideals; principal ideals, operations on ideals. [4]

4.The quotient construction (for rings): the construction and connection with homomorphisms; maximal ideals; ideals of the quotient ring; examples. [3]

5.Polynomial rings and unique factorisation: polynomial rings; division algorithm; unique factorisation. [3]

6.Constructing roots of polynomials: construction of extension fields; examples, including finite fields. [4]

J.B. Fraleigh, A First Course in Abstract Algebra, (any edition: the library has many copies) Addison-Wesley (recommended but not essential).

Also similar books like:

R.B.J.T. Allenby, Rings, Fields and Groups: an Introduction to Abstract Algebra, Addison-Wesley.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Louise Walker - Unit coordinator