MATH30130 Number Theory

Academic Year 2021/2022

This course will be an introduction to algebraic number theory. Topics will be chosen from: Basic Structures in Algebraic Number Theory (Euclidean Domains, PIDs, UFDs, Fundamental Theorem of Arithmetic for certain rings), Fermat's Infinite Descent, Fermat's Last Theorem for small exponents, Gauss' reciprocity law and classification of conics over finite fields, point counting over finite fields, p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma, conics over p-adic fields and Hilbert symbol, Hasse-Minkowski theorem and classification of conics over number fields.

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Curricular information is subject to change

Learning Outcomes:

The student expected develop an understanding of the following concepts:

(i) Basic structures in Algebraic Number Theory (Euclidean Domains, PIDs, UFDs, Fundamental Theorem of Arithmetic for certain rings)

(ii) Fermat's method of infinite descent

(iii) Proof of Fermat's Last Theorem for small exponents based on field arithmetic

(iv) Gauss' reciprocity law and classification of conics over finite fields

(v) Point counting over finite fields (Chevalley-Warning Theorem).

(vi) p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma, conics over p-adic fields and Hilbert symbol.

(vii) Hasse-Minkowski theorem and classification of conics over number fields.

Indicative Module Content:

Basic structures in Algebraic Number Theory (Euclidean Domains, PIDs, UFDs, Fundamental Theorem of Arithmetic for certain rings), Gauss' reciprocity law, Chevalley-Warning Theorem, p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma, conics over p-adic fields and Hilbert symbol, Hasse-Minkowski theorem and classification of conics over number fields.

Student Effort Hours: 
Student Effort Type Hours
Lectures

30

Tutorial

6

Specified Learning Activities

24

Autonomous Student Learning

48

Total

108

Approaches to Teaching and Learning:
Active/Task-based Learning
Lectures
Enquiry & Problem-based Learning 
Requirements, Exclusions and Recommendations
Learning Requirements:

Contents of MATH10040 Numbers and Functions (mathematical induction; solutions of inequalities; basic concepts associated with function; greatest common divisors; theorems of Fermat and Euler; basic concepts of Number Theory, such as prime numbers and their relevance to codes)

Contents of MATH20310 Groups, Rings and Fields (definition and examples of groups, subgroups, cosets and Lagrange's Theorem, the order of an element of a group, normal subgroups and quotient groups, group homomorphisms and the isomorphism theorems, definitions of a commutative ring with unity, integral domains and fields, units, irreducibles and primes in a ring, ideals and quotient rings, prime and maximal ideals, ring homomorphisms and the homomorphism theorem, polynomial rings, the division algorithm, gcd for polynomials, irreducible polynomials and field extensions)


Module Requisites and Incompatibles
Required:
MATH10040 - Numbers & Functions, MATH20310 - Groups, Rings and Fields


 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: Final Exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

60

Examination: Midterm Examination (tentatively: to be scheduled on Week 8) Varies over the Trimester No Standard conversion grade scale 40% No

20

Assignment: A total of 4 homework sets Varies over the Trimester n/a Standard conversion grade scale 40% No

20


Carry forward of passed components
No
 
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Self-assessment activities

How will my Feedback be Delivered?

Self-assessment activities are tentative and will be based on the following model: Students will be asked to grade their own responses to select assignments, based on a detailed solution sheet. They will prepare a detailed report on their weaknesses on key aspects of the module. Based on the accuracy of this report, their homework grades will be positively updated. The other two feedback strategies to employed (Feedback individually to students, post-assessment and Group/class feedback, post-assessment) are self-explanatory.

Title: Number Theory 1: Fermat's Dream (Translations of Mathematical Monographs) (Vol 1)
Authors: Kazuya Kato, Nobushige Kurokawa, Takeshi Saito

Title: Number Theory (Pure and Applied Mathematics Book 20)
Authors: Z. I. Borevich and I. R. Shafarevich
Name Role
Dr Daniele Casazza Lecturer / Co-Lecturer