Module manager: Professor P Martin
Email: ppmartin@maths.leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2009/10
| MATH1022 | Introductory Group Theory |
| MATH3171 | Algebra and Numbers |
This module is approved as an Elective
A ring is an algebraic system in which addition, subtraction and multiplication may be performed. Integers, polynomials and matrices all provide examples of rings, so this notion covers an important range of mathematical structures. They are studied in this module. One topic is the generalization to some other rings of the Fundamental Theorem of Arithmetic, that every positive integer can be written in a unique way as a product of primes. Another topic is Kronecker's theorem that every non-constant polynomial with coefficients in a field has a root in some larger field. Combined, one can begin to understand the possible finite fields.
On completion of this module, students should be able to:
a) Define some of the main concepts about rings, polynomials and fields;
b) State and prove some of the basic results about rings, polynomials and fields;
c) State the axioms of a ring and deduce basic properties directly from them;
d) Identify subrings, ideals and units in the main examples of rings;
e) Use the First Isomorphism Theorem to exhibit isomorphisms between rings;
f) Demonstrate understanding of unique factorisation or lack of it;
g) Identify irreducibles in various examples of rings, using appropriate tests;
h) Make computations in fields obtained by adjoining an algebraic element.
1. Rings, fields and integral domains. Units. Subrings. Examples, such as the ring obtained by adjoining a square root to the integers, matrix rings, polynomial rings and the field of fractions of an integral domain.
2. Ideals, factor rings, homomorphisms and isomorphisms. Principal ideal domains. The main examples of principal ideal domains. The First Isomorphism Theorem for rings.
3. Factorization in integral domains. The notion of a greatest common divisor (at least in a principal ideal domain). Euclid's Algorithm. Prime and irreducible elements in integral domains. Any principal ideal domain is a unique factorization domain. Fundamental Theorem of Arithmetic. Factorization of polynomials. Gauss' Lemma. Eisenstein's criterion. The rational root test.
4. Field extensions. Minimal polynomials of algebraic elements in field extensions. Structure of the field obtained by adjoining an algebraic element. Kronecker's Theorem. The characteristic of a field and the prime subfield. The number of elements in a finite field is a prime power. Examples of finite fields.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Example Class | 11 | 1 | 11 |
| Lecture | 22 | 1 | 22 |
| Private study hours | 67 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 100 | ||
Examples sheets.
| Exam type | Exam duration | % of formal assessment |
|---|---|---|
| Standard exam (closed essays, MCQs etc) | 2.0 Hrs Mins | 100 |
| Total percentage (Assessment Exams) | 100 | |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
The reading list is available from the Library website
Last updated: 5/24/2010
Errors, omissions, failed links etc should be notified to the Catalogue Team