MA257: Introduction to Number Theory

Lecturer Lecture times Lecture room
Prof John Cremona
Zeeman C2.21 (ext 24403) 		Monday 11-12 L3
Tuesday 11-12 Arts Cinema
Wednesday 9-10 Arts Cinema
Teaching assistants Support class time Support class room
Florian Bouyer Monday 12-1 A0.23 (Social Sciences)
Angelos Koutsianas Tuesday 4-5 H0.52 (Humanities, ground floor)
Chris Birkbeck Friday 1-2 B2.01 (Science Concourse)

Lecture Log for 2015

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 (only the Introduction and Chapter 1 are currently available). Each chapter heading is a link to the larger format version (without proofs) used in the cinema lectures.

Next lecture

Lecture Date Material covered Notes
1 5 Jan INTRODUCTION: What is Number Theory? 1-3
2-6   CHAPTER ONE: Factorization 4-13
2 6 Jan Review of divisibility, gcd and Bezout. 4-6
3 7 Jan Review of Euclidean Algorithm (EA) in $\mathbb{Z}$, primes and unique factorization in $\mathbb{Z}$ and $\mathbb{Q}$.
Introduction to Euclidean Domains (EDs) and Unique Factorization Domains (UFDs). 		7-8
4 12 Jan The Gaussian integers $\mathbb{Z}[i]$ as a Euclidean Domain, PID and UFD. Euclidean implies PID. Gaussian norm and units. 9-11
5 13 Jan gcds, primes and irreducibles in a PID. Examples from $\mathbb{Z}[i]$. Existence of irreducible factors in a PID. 11-13
6 14 Jan PID implies UFD. Gaussian primes. Mention other examples of UFDs and non-UFDs. 11-13
7-11   CHAPTER TWO: Congruences and modular arithmetic 14-23
7 19 Jan Congruence. Solving linear congruences. Residue classes. 14-15
8 20 Jan The ring $\mathbb{Z}/m\mathbb{Z}$ and group $U_m$. Euler's $\varphi$-function. Fermat's and Euler's Theorems. 16-18
9 21 Jan Wilson's Theorem. Applications. 18-19
10 26 Jan Infinitely many primes $\equiv1\pmod{m}$ for $m=4, 8, 16, \dots, q$.
Chinese Remainder Theorem and applications. 		19-21
11 27 Jan Formula for $\varphi(m)$. Structure of $U_m$: primitive roots for primes. 21-22
12a 28 Jan Primitive roots: the general case. Examples. 22-23
12-14   CHAPTER THREE: Quadratic Reciprocity 24-28
12b 28 Jan Quadratic residues and non-residues: definition and first examples. 24
13 2 Feb Legendre symbols. Euler's criterion. Quadratic reciprocity: Gauss's Lemma. 25-26
14 3 Feb Quadratic reciprocity: quadratic character of 2, full statement, examples. 27-28
15-19   CHAPTER FOUR: Diophantine Equations 29-36
15 4 Feb Geometry of Numbers: Minkowski's Theorem and sums of 2 squares. 29-30
16 9 Feb Sums of 3 and 4 squares. 30-31
17 10 Feb Legendre's Theorem. 31-33
18 11 Feb Blichfeldt's Theorem, proof of Minkowski's Theorem. 36-37
19 16 Feb Pythagorean Triples, Fermat's Last Theorem for exponent 4. 33-36
20-27   CHAPTER FIVE: p-adic numbers 38-48
20 17 Feb Introduction to $p$-adic numbers, examples. Definition of $\mathbb{Z}_p$. 38-39
21 18 Feb Arithmetic in $\mathbb{Z}_p$:commutative ring, no zero-divisors, characterization of units. 39-41
22 23 Feb Divisibility and unique factorization in $\mathbb{Z}_p$. 41-42
23 24 Feb Intersection with $\mathbb{Q}$. The $\mbox{ord}_p$ function on $\mathbb{Z}_p$. 42-43
24 25 Feb The field $\mathbb{Q}_p$. Normed fields. The $p$-adic norm on $\mathbb{Q}$ and $\mathbb{Q}_p$. 43-44
25 2 Mar Convergence in $\mathbb{Q}_p$. $\mathbb{Q}_p$ as a completion of $\mathbb{Q}$. Numerical examples of convergent sequences and series. 44-45
26 3 Mar Squares in $\mathbb{Z}_p$ and $\mathbb{Q}_p$. 46-47
27 4 Mar Solving equations in $\mathbb{Q}_p$; Hensel lifting. 47-49
28-30   CHAPTER SIX: additional topics  
28 9 Mar Perfect numbers and Mersenne primes.  
29 10 Mar Primality tests and proving primality.  
30 11 Mar Integer factorization. Revision questions (on request).  
Lectures have finished.

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