Algebra and Number Theory
| Code | School | Level | Credits | Semesters |
| MATH2103 | Mathematical Sciences | 2 | 20 | Full Year UK |
- Code
- MATH2103
- School
- Mathematical Sciences
- Level
- 2
- Credits
- 20
- Semesters
- Full Year UK
Summary
Building upon first year Core Mathematics and further basic pure mathematics concepts introduced in Algebra, this course will develop in more detail the fundamental concepts in algebra, introduce elementary number theory, and explore the interaction of the two subjects. This interplay has been a rich source of fascinating mathematics and has led to important breakthroughs, including the proof of Fermat’s Last Theorem.
Topics include:
Algebra:
Basic concepts concerning rings will be developed, in particular:
- Substructures (including subrings, ideals) and quotient structures.
- Structure preserving maps between algebraic objects (including ring homomorphisms).
- Application of algebraic concepts to geometry and number theory.
Number Theory:
Focus is on classical problems, including:
- Primes and integer factorisation.
- Modular arithmetic.
- Fermat’s Little Theorem, its application to primality testing, and methods of factorisation.
Target Students
Single Honours and Joint Honours students from the School of Mathematical Sciences.
Classes
- One 2-hour lecture
- One 1-hour lecture
Teaching will be through a variety of methods, including traditional lectures and problem classes, with the delivery tailored to the material on a week-by-week basis.
Assessment
- 40% Coursework: Summative assessment based on tasks distributed through the year.
- 60% Exam (2-hour): Written examination – Spring.
Assessed by end of spring semester
Educational Aims
This course is a key component in Pure Mathematics, being a pre-requisite to all further courses in algebra and number theory at level 3 and level 4. The course aims to provide students with a thorough grounding in the fundamental concepts of algebra and number theory. Students will learn how theories and techniques in these areas are developed starting from basic principles and using rigorous methods of proof. They will learn how to apply these theories and techniques to solve problems in these areas. Students will also learn how general algebraic principles can be applied in the different contexts of geometry and number theory. Throughout the course students will acquire skills and knowledge that prepares them for the more advanced and more specialised theories in algebra, geometry and number theory that are taught at level 3 and 4.Learning Outcomes
A student who completes this course successfully will be able to:
L1 – Demonstrate knowledge and understanding of the main concepts and theorems of algebra and number theory.
L2 – Reason logically and work analytically to construct rigorous mathematical proofs of standard and basic (unseen) propositions in algebra and number theory.
L3 – Develop and justify a mathematical framework using key definitions and theorems from algebra and number theory to solve given routine and unseen problems to a high level of accuracy.
L4 – Apply and illustrate the main concepts and theorems of algebra through the exploration of examples, including those from geometry and number theory.
L5 – Present conclusions in writing using structured and mathematically rigorous arguments and contextually appropriate language.
L6 – Summarise complex ideas clearly and concisely, taking due consideration of the target audience.