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### Course Overview

Many examples of rings will be familiar before entering this course. Examples include the integers modulo n, the complex numbers and n-by-n matrices with real entries. The course develops from the fundamental definition of ring to study particular classes of rings and how they relate to each other. We also encounter generalisations of familiar concepts, such as what is means for a polynomial to be prime.

### Course Details

Study Type Level Undergraduate 3 Second Sub Session 15 credits (7.5 ECTS credits) None. No Dr Zur Izhakian

### Qualification Prerequisites

• Either Programme Level 3 or Programme Level 4

### What courses & programmes must have been taken before this course?

• MX3020 Group Theory (Studied)
• Either MA2008 Linear Algebra I (Passed) or MA2506 Linear Algebra (Passed)

None.

None.

No

### Course Description

• Basic concepts and examples. Ideals, factor rings, isomorphism theorems.
• Rings of polynomials.
• Field of fractions of a domain.
• Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains.
• Passage from R to R[X]. Gauss's Theorem. Eisenstein's criterion.
• Fields : characteristic, prime subfield.
• Finite fields, construction.
• Algebraic and transcendental elements, algebraic closure.

Syllabus

• Rings
• Zero divisors and integral domains.
• Homomorphisms.
• Ideals and quotient rings.
• Field of fractions of an ID.
• ID, UFD, PID and ED.
• Polynomial rings over commutative rings.
• Field extensions
• Splitting fields.
• Finite fields.

### Degree Programmes for which this Course is Prescribed

• BSc Education (Secondary) - Mathematics
• BSc Mathematics
• BSc Mathematics with Gaelic
• MA Mathematics
• Mathematics Joint
• Mathematics Major

### Contact Teaching Time

56 hours

This is the total time spent in lectures, tutorials and other class teaching.

### Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment). Only the marks obtained on first sitting can be used for Honours classification.

### Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

### Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.

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