*MATH3078 *Further Number Theory

## Module Overview

Number Theory is the study of integers and their generalisations such as the rationals, algebraic integers or finite fields. The problem more or less defining Number Theory is to find integer solutions to equations, such as the famous Fermat equation x^n + y^n = z^n. In this module we build on the group, ring and number theoretic foundations laid in MATH1001, MATH2003 and MATH3086. We will first prove a structure theorem for the group of units modulo n. We then move on to the famous Gaussian Quadratic Reciprocity Law which yields an algorithm to decide solvability of quadratic equations over finite fields. Using geometric as well as algebraic methods, we will then characterise which integers can be written as the sum of two and four squares, respectively. The former leads us naturally to the study of binary quadratic forms, a central topic of this module. In the final part of this module, we will explore rings of integers in algebraic number fields; they generalise the role the integers play within the rational numbers; the simplest new example is the ring of Gaussian integers, Z[i]. We will investigate to what extent certain central properties of the integers, such as unique prime power factorisation, generalises to these rings. The deviation from unique prime factorisation is measured by the so-called ideal class group, probably the most important invariant of algebraic number fields. It can be seen that it is finite and that its order for quadratic number fields is intimately related to the number of equivalence classes of quadratic forms introduced earlier in the module.

### Aims and Objectives

#### Module Aims

We will study binary integral quadratic forms and rings of integers in algebraic number fields and encounter an astounding relationship between these two concepts.

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

- techniques to study quadratic congruences
- techniques to study quadratic congruences
- techniques to study quadratic congruences
- techniques to study quadratic congruences
- techniques to study quadratic congruences
- fundamental concepts in the theory of binary quadratic forms
- fundamental concepts in the theory of binary quadratic forms
- fundamental concepts in the theory of binary quadratic forms
- fundamental concepts in the theory of binary quadratic forms
- fundamental concepts in the theory of binary quadratic forms
- basic aspects of the theory of algebraic number fields and their rings of integers
- basic aspects of the theory of algebraic number fields and their rings of integers
- basic aspects of the theory of algebraic number fields and their rings of integers
- basic aspects of the theory of algebraic number fields and their rings of integers
- basic aspects of the theory of algebraic number fields and their rings of integers

##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

- do abstract, analytical and structured thinking
- do abstract, analytical and structured thinking
- do abstract, analytical and structured thinking
- do abstract, analytical and structured thinking
- do abstract, analytical and structured thinking

##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

- understand and compose rigorous mathematical proofs
- understand and compose rigorous mathematical proofs
- understand and compose rigorous mathematical proofs
- understand and compose rigorous mathematical proofs
- understand and compose rigorous mathematical proofs

##### Learning Outcomes

Having successfully completed this module you will be able to:

- apply techniques to study quadratic congruences.
- work with the fundamental concepts in the theory of binary quadratic forms.
- work with the basic concepts of the theory of algebraic number fields and their rings of integers.

### Syllabus

Quadratic congruences: • group of units modulo n • quadratic residues and the Legendre symbol • Euler's criterion, Gauss' lemma • quadratic reciprocity law Binary quadratic forms: • integers as sums of two and four squares, Minkowski's lattice point theorem • irreducible elements in the Gaussian integers • equivalence of binary quadratic forms, reduced quadratic forms, finiteness of the class number Algebraic number theory: • algebraic numbers and rings of integers • trace and norm • quadratic and cyclotomic number fields • (non-)unique factorization into irreducibles • ideals in rings of integers, ideal class group, finiteness of the class number • ideal classes in quadratic number fields and equivalence classes of binary quadratic forms

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem sheets, private study

Type | Hours |
---|---|

Independent Study | 102 |

Teaching | 48 |

Total study time | 150 |

#### Resources & Reading list

R A Mollin (1999). Algebraic Number Theory.

I N Stewart, D O Tall (2002). Algebraic Number Theory and Fermat's Last Theorem.

R A Mollin (1999). Algebraic Number Theory.

D Zagier (1981). Zetafunktionen und quadratische Koerper.

I N Stewart, D O Tall (2002). Algebraic Number Theory and Fermat's Last Theorem.

H Davenport (1992). The Higher Arithmetic.

R A Mollin (1999). Algebraic Number Theory.

D Zagier (1981). Zetafunktionen und quadratische Koerper.

R A Mollin (1999). Algebraic Number Theory.

I N Stewart, D O Tall (2002). Algebraic Number Theory and Fermat's Last Theorem.

G A Jones, J M Jones (1998). Elementary Number Theory.

G A Jones, J M Jones (1998). Elementary Number Theory.

H Davenport (1992). The Higher Arithmetic.

D Zagier (1981). Zetafunktionen und quadratische Koerper.

G A Jones, J M Jones (1998). Elementary Number Theory.

R A Mollin (1999). Algebraic Number Theory.

D Zagier (1981). Zetafunktionen und quadratische Koerper.

H Davenport (1992). The Higher Arithmetic.

I N Stewart, D O Tall (2002). Algebraic Number Theory and Fermat's Last Theorem.

D Zagier (1981). Zetafunktionen und quadratische Koerper.

G A Jones, J M Jones (1998). Elementary Number Theory.

H Davenport (1992). The Higher Arithmetic.

I N Stewart, D O Tall (2002). Algebraic Number Theory and Fermat's Last Theorem.

H Davenport (1992). The Higher Arithmetic.

G A Jones, J M Jones (1998). Elementary Number Theory.

### Assessment

#### Summative

Method | Percentage contribution |
---|---|

Coursework | 20% |

Exam (120 minutes) | 80% |

#### Referral

Method | Percentage contribution |
---|---|

Exam (120 minutes) | 100% |

#### Repeat Information

**Repeat type: Internal & External**

### Linked modules

Prerequisites: MATH1001 AND MATH3086