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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 Ehud Meir

### Qualification Prerequisites

• Either Programme Level 3 or Programme Level 4

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### 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.

### Contact Teaching Time

Information on contact teaching time is available from the course guide.

### Teaching Breakdown

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

### Summative Assessments

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).

### 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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