Undergraduate Course: Introduction to Number Theory (MATH10071)
Course Outline
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 
SCQF Credits  10 
ECTS Credits  5 
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. 
Course description 
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 coprime with n.

Information for Visiting Students
Prerequisites  Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling. 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2023/24, Available to all students (SV1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)  Introduction to Number Theory (MATH10071)  2:00  
Learning Outcomes
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, EulerÂ¿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

Reading List
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 Lokeng Hua, SpringerVerlag, 1982. 
Additional Information
Graduate Attributes and Skills 
Not entered 
Study Abroad 
Not Applicable. 
Keywords  INT 
Contacts
Course organiser  Dr Agata Smoktunowicz
Tel:
Email: 
Course secretary  Miss Greta Mazelyte
Tel:
Email: 

