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MX4545: NUMBER THEORY (2017-2018)

Last modified: 25 May 2018 11:16


Course Overview

This course concerns the integers, and more generally the ring of algebraic integers in an algebraic number field.  The course begins with statements concerning the rational integers, for example we discuss the Legendre symbol and quadratic reciprocity. We also prove a result concerning the distribution of prime numbers. In the latter part of the course we study the ring of algebraic integers in an algebraic number field. One crucial result is the unique factorisation of a nonzero ideal as a product of primes, generalising classical prime factorisation in the integers.



Course Details

Study Type Undergraduate Level 4
Session Second Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus None. Sustained Study No
Co-ordinators
  • Dr Lubna Shaheen

Qualification Prerequisites

  • Programme Level 4

What courses & programmes must have been taken before this course?

  • MX4082 Galois Theory (Studied)
  • MX3531 Rings & Fields (Passed)
  • Any Undergraduate Programme (Studied)

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

Number theory is the study of integers and has three main branches: Elementary, Analytical and Algebraic. This course consists of a selection of topics from these branches. The topics will include some of the following: the theory of quadratic congruences, continued fractions, pseudo-primes, primitive roots, Diophantine equations, the distribution of prime numbers, algebraic integers in quadratic number fields.

 

Syllabus

  • Polynomial Congruences.
  • Quadratic Residues and Quadratic Reciprocity.
  • Bertrand’s postulate.
  • Algebraic number fields.
  • Algebraic integers.
  • Quadratic number fields and their rings of algebraic integers.
  • Cyclotomic number fields and their rings of algebraic integers.
  • The factorisation of ideals in rings of algebraic integers.

Further Information & Notes

Course Aims

The overall aim of the course is to study elementary and algebraic number theory, making use of
methods from group theory, ring theory and Galois theory. The topics covered will be
- Polynomial Congruences. 
- Quadratic Residues and Quadratic Reciprocity. 
- Bertrand’s postulate.
- Algebraic number fields. 
- Algebraic integers.
- Quadratic number fields and their rings of algebraic integers.
- Cyclotomic number fields and their rings of algebraic integers.
- The factorisation of ideals in rings of algebraic integers.

 

Learning Objectives
By the end of the course the student should:
- be able to solve congruences of small degrees;
- be able to evaluate the Legendre symbol and to make use of its applications;
- be able to prove the existence of infinitely many primes in certain arithmetic progressions;
- be able to understand and apply the factorisation theory of quadratic integers;
- be able to factorise ideals in rings of algebraic integers.

Degree Programmes for which this Course is Prescribed

None.

Contact Teaching Time

32 hours

This is the total time spent in lectures, tutorials and other class teaching.

Teaching Breakdown


Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).
Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).
Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination. Students undertake practice questions in tutorials allowing formative assessment and feedback from tutors.

Feedback

In-course assessment will be marked and feedback provided to the students.

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