| Lecturer | Lecture times | Lecture room |
|---|---|---|
| Prof John Cremona Zeeman C2.21 (ext 24403) |
Monday 11-12 | L3 |
| Tuesday 11-12 | OC0.03 | |
| Wednesday 9-10 | OC0.03 | |
| Teaching assistants | Support class time | Support class room |
| Marco Caselli | Monday 12-1 | S0.18 |
| George Turcas | Tuesday 4-5 | S0.18 |
| Mattia Sanna | Friday 11-12 | Week 2: A1.01 Zeeman |
| Weeks 3,4: D1.07 Zeeman | ||
| Week 5: Arts Cinema | ||
| Weeks 6-10: OC1.04 |
The material covered in each lecture will appear here throughout the term.
In the right columns are the approximate relevant pages in the full lecture notes. Each chapter heading is a link to the larger format version (without proofs) used in lectures.
An alternative experimental web version of the lecture notes is available (thanks to David Farmer).
| Lecture | Date | Material covered | Notes | |
|---|---|---|---|---|
| 1 | 8 Jan | INTRODUCTION: What is Number Theory? | 1-3 | |
| 2-6 | CHAPTER ONE: Factorization | 4-13 | ||
| 2 | 9 Jan |
(1.1) Review of divisibility, "to contain is to divide", $\mathbb{Z}$
as a PID.
(1.2) Review of integer gcd; Bezout identity. |
4-6 | |
| 3 | 10 Jan |
(1.3) Review of Euclidean Algorithm (EA) in $\mathbb{Z}$.
(1.4) Review of primes and unique factorization in $\mathbb{Z}$ and $\mathbb{Q}$. |
7-8 | |
| 4 | 15 Jan |
(1.5) Introduction to Euclidean Domains (EDs) and Unique Factorization Domains (UFDs).
The Gaussian integers $\mathbb{Z}[i]$ as a Euclidean Domain, PID and UFD. Euclidean implies PID. Gaussian norm and units. |
9-11 | |
| 5 | 16 Jan | gcds, primes and irreducibles in a PID. | 11-13 | |
| 6 | 17 Jan | Existence of irreducible factors and factorizations in a PID. PID implies UFD. Gaussian primes and factorization examples. | 11-13 | |
| 7-11 | CHAPTER TWO: Congruences and modular arithmetic | 14-23 | ||
| 7 | 22 Jan | Congruence. Solving linear congruences. Residue classes. | 14-15 | |
| 8 | 23 Jan | The ring $\mathbb{Z}/m\mathbb{Z}$ The unit group $U_m$ of $\mathbb{Z}/m\mathbb{Z}$. Euler's $\varphi$-function. The field $\mathbb{F}_p$. | 16-18 | |
| 9 | 24 Jan | Fermat's and Euler's Theorems. Wilson's Theorem. Applications of Fermat and Euler. Solvability of \(x^2\equiv-1\pmod{p}\). | 18-19 | |
| 10 | 29 Jan | More applications: Primes $p\equiv1\pmod4$ are sums of 2 squares. Infinitely many primes $\equiv1\pmod{m}$ for $m=4, 8, 16, \dots, q$. | 19-21 | |
| 11 | 30 Jan | Chinese Remainder Theorem and applications. Multiplicativity of $\varphi$, formula for $\varphi(m)$. | 21-22 | |
| 12 | 31 Jan | CRT examples. Primitive roots. | 22-23 | |
| 13a | 5 Feb | Existence / nonexistence of primitive roots for composite moduli. | 23 | |
| 13-15 | CHAPTER THREE: Quadratic Reciprocity | 23-27 | ||
| 14 | 6 Feb | Quadratic residues and non-residues: definition and first examples. Legendre symbols. Euler's criterion. Gauss's Lemma (statement). | 23-24 | |
| 15 | 7 Feb | Gauss's Lemma (proof). Quadratic character of 2. Law of Quadratic Reciprocity. | 25-27 | |
| 16a | 12 Feb | Quadratic Reciprocity examples. | 27 | |
| 16-19 | CHAPTER FOUR: Diophantine Equations | 29-36 | ||
| 16b | 12 Feb | Geometry of Numbers: introductory examples, lattices. | 30-31 | |
| 17 | 13 Feb |
Minkowski's Theorem, sums of 2 squares. Sums of 3 squares.
Sums of 4 squares (start). |
31-33 | |
| 18 | 14 Feb |
Sums of 4 squares (end).
Legendre's Theorem (start). |
36-37 | |
| 19 | 19 Feb | Legendre's Theorem (proof). Example. | 33-36 | |
| 20 | 20 Feb | Pythagorean Triples. Fermat's Last Theorem for exponent 4. | 38-39 | |
| 20-27 | CHAPTER FIVE: p-adic numbers | 38-48 | ||
| 21 | 21 Feb | Introduction to $p$-adic numbers, examples. Definition of $\mathbb{Z}_p$. | 39-41 | |
| 22 | 26 Feb | Arithmetic in $\mathbb{Z}_p$:commutative ring, no zero-divisors, characterization of units. $p$-adically integral rationals. | 41-42 | |
| 23 | 27 Feb |
Divisibility and unique factorization in $\mathbb{Z}_p$.
The $\mbox{ord}_p$ function on $\mathbb{Z}_p$. |
42-43 | |
| 24 | 28 Feb |
The field $\mathbb{Q}_p$. Example calculations in $\mathbb{Q}_p$.
Introduction to normed fields and the $p$-adic norm on $\mathbb{Q}$ and $\mathbb{Q}_p$. |
43-44 | |
| 25 | 5 Mar | Normed fields and the $p$-adic norm on $\mathbb{Q}$ and $\mathbb{Q}_p$. Convergence in $\mathbb{Q}_p$. | 44-45 | |
| 26 | 6 Mar |
$\mathbb{Q}_p$ as a completion of $\mathbb{Q}$.
Numerical examples of convergent $p$-adic sequences and series.
Squares in $\mathbb{Q}_p$ (for $p$ odd). |
46-47 | |
| 27 | 7 Mar |
Squares in $\mathbb{Q}_2$.
Solving equations in $\mathbb{Q}_p$: Hensel lifting. Examples. |
47-48 | |
| 28-30 | CHAPTER SIX: additional topics (not examinable) | |||
| 28 | 12 Mar | Mersenne numbers: Mersenne primes, perfect numbers, the Lucas-Lehmer test. | ||
| 29 | 13 Mar | Primality tests, pseudoprimes and strong pseudoprimes, Miller-Rabin test. | ||
| 30 | 14 Mar |
Proving primality: Pocklington-Lehmer test. Fast modular exponentiation.
Assorted revision examples. |
||
| Lectures have finished. | ||||