Undergraduate Course: Introduction to Number Theory (MATH10071)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
||Availability||Available to all students
|Summary||This is a level 10, Year 3 course with prerequisite Fundamentals of Pure Mathematics. We assume familiarity with the basics of prime numbers, unique factorisation of integers and modular arithmetic. The main theme of the course will be the interplay between Number Theory and rings.
1. Binary operations on integers, axioms of a ring.
2. The ring of integers Z. The division algorithm. Euclidean algorithm. Primes, units, irreducibles in Z. Irreducibles in Z are primes in Z. Factorisation in integers. The Fundamental Theorem of Arithmetic.
3. Rings. Axioms of a ring. Deducing some basic properties from axioms, for example deducing that the zero element in any ring is unique. Using rings to prove some basic results from elementary number theory.
5. Integral domains, zero divisors. Cancellation in domains. Greatest common divisor.
6. Gaussian integers and rings Z[d] where d is some irrational number. Units, primes, irreducibles in such rings. Using properties of the ring of Gaussian integers to determine which integers can be written as sums of squares of two integers.
7. Ideals in rings. Factor rings. Examples of rings.
8. Euclidean domains. Uniqueness of factorisation. Primes and irreducibles in Euclidean domains. Euclidean algorithm for Euclidean domains. The ring of Gaussian integers is a Euclidean domain.
9. Connections of Gaussian integers and quadratic residues and the Legandre symbol.
10. Exercises on the Legandre symbol and quadratic residues. We will solve a variety of questions on quadratic residues using the five basic rules of calculating quadratic residues. We will assume some results such as Gauss Lemma and the Law of Quadratic residues without proofs and will concentrate on being able to use them for calculating the Legendre symbol in a variety of exercises. We will mention surprising connections of the Legendre symbol and Gaussian integers.
11. Applications of previous material to linear and quadratic congruences. Linear and quadratic congruences. We will apply the obtained results on Legandre symbol to determine how many integer solutions have some quadratic congruences. We will apply the Euclidean algorithm to solve linear congruences.
12. Euler's function. We will apply the formula for Euler's function in the solving of a variety of exercises. Let n be an integer larger than 0. The Euler's function gives the number of integers which are larger than zero and not exceeding n and are co-prime with n.
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Academic year 2023/24, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 10%, Examination 90%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Introduction to Number Theory (MATH10071)||2:00|
On completion of this course, the student will be able to:
- Solve linear and quadratic congruences.
- Know basic results in elementary number theory, including quadratic residues, Gaussian integers, Legendre symbol, Euler¿s theorem.
- Demonstrate familiarity with methods for writing an integer as a sum of two squares.
- Appreciate some algebraic techniques in number theory.
- Know some modern applications of number theory, mainly know basics of coding theory
|Elementary Number Theory, by Kenneth H. Rosen, 6th Edition, 2010, Pearson.|
- A friendly introduction to number theory by J. H. Silverman, Prentice Hall, 2001.
- Introduction to number theory by Lo-keng Hua, Springer-Verlag, 1982.
|Graduate Attributes and Skills
|Course organiser||Dr Agata Smoktunowicz
|Course secretary||Miss Greta Mazelyte