Introduction to Algebraic Number Theory Part I

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1 Introduction to Algebraic Number Theory Part I A. S. Mosunov University of Waterloo Math Circles November 7th, 2018

2 Goals Explore the area of mathematics called Algebraic Number Theory. Specifically, we will see how to generalize the notions of integers, rational numbers, prime numbers, etc. Goal 1. Understand the basics of the theory. Goal 2. See beautiful theorems. Goal 3. Understand open problems.

3 Number Theories Number theory studies properties of numbers, such as 2, 1,22/7, 2, or π. There are many subareas of number theory, such as Analytic number theory, Theory of Diophantine approximation, etc. Algebraic number theory studies numbers that are roots of polynomial equations, such as 3, which is a root of x + 3 = 0, 2, which is a root of x 2 2 = 0, i, which is a root of x = 0. Transcendental number theory studies numbers that do not satisfy this property, such as π, log 2 or 2 2. Determining whether a number is algebraic or transcendental can be very hard! Is π transcendental?

4 Why Study Number Theory? Figure: Messaging apps that (hopefully!) use cryptographic protocols based on hard number theoretical problems

5 Why Study Number Theory? It is beautiful. It is applicable! Many cryptographic protocols reside on difficult number theoretical problems. Many protocols, such as RSA or the Diffie-Hellman Protocol, which are based on regular number theory are vulnerable to quantum computer attacks. Algebraic number theory comes to the rescue! Lattice-based cryptography is quantum-safe and it uses properties of numbers that are roots of x n + 1 = 0. CSIDH is a cryptographic protocol that is quantum-safe and it uses properties of numbers of the form a + b m, where m is a very small negative integer and a, b are rational numbers.

6 BACKGROUND

7 Rational Integers The numbers..., 2, 1,0,1,2,... are called (rational) integers. The set of all integers is denoted by Z. Let a and b be integers. We say that a divides b when b = ak for some integer k. We write a b in this case, and a b otherwise. A number p 2 is a (rational) prime if it is divisible only by 1 and p. The Fundamental Theorem of Arithmetic. Any integer greater than 1 can be written uniquely (up to reordering) as the product of primes. Let a and b be integers. The largest integer g such that g a and g b is called the greatest common divisor of a and b. It is denoted by gcd(a,b). The numbers a and b are called coprime if gcd(a,b) = 1.

8 Detour: Rational Numbers and Apéry s Theorem A number is called rational if it is of the form a/b for some rational integers a and b, where a 1. The set of all rational numbers is denoted by Q. Determining whether a given number is rational or irrational can be very hard! In 1979 the French mathematician Roger Apéry proved that the number ζ (3) = is irrational. It is still unknown whether ζ (5), ζ (7),ζ (9) or ζ (11) are irrational. However, at least one of them is (proved by Wadim Zudilin in the 90 s). See the article A proof that Euler missed by Alfred van der Poorten: //

9 Detour: Rational Numbers and Apéry s Theorem Figure: Roger Apéry ( )

10 Exercise If a,b are coprime positive integers and ab = c 2 for some integer c, show that a = t 2 and b = s 2 for some integers t and s. Show that for any integer x the numbers x and x are coprime. Numbers 0,1,2 2 = 4,3 2 = 9,... are called squares. Show that the distance between k 2 and (k + 1) 2 is equal to 2k + 1. When is this distance equal to 1? Use the previous results to conclude that the equation y 2 = x 3 + x has no solutions in positive integers x and y.

11 ALGEBRAIC NUMBER THEORY BEGINS

12 How Euler Almost Discovered Algebraic NT Can the distance between a square and a cube be equal to one? In 1700 s, Euler showed that the only square and cube that differ by 1 are 8 and 9. Homework. Prove that the equation y 2 = x has only one solution in positive integers. Hint: use the fact that (x 3 + 1) = (x + 1)(x 2 x + 1). He also almost proved that the only square and cube that differ by 2 are 25 and 27. Idea: consider the equation y 2 = x 3 2 and write it as (y + 2)(y 2) = x 3. If y + 2 and y 2 are coprime, they must be cubes. But what does coprime even mean in this setting?

13 Detour: Theorems of Mordell and Tijdeman We have already seen that the distance between consecutive squares grows. Same observation applies to cubes. Does the distance between consecutive squares and cubes grow? 0,1,4,8,9,16,25,27,36,49,64,81,100,121,125,144,... The answer is yes. This was proved by the British mathematician Loius Mordell in 1960 s. In 1976, Robert Tijdeman showed that the number of consecutive powers that differ by 1 is finite. Questions about larger distances is still open. The solutions (x,y,m,n) to the equation y m = x n + 1 must satisfy x, y,m,n e eee730.

14 Detour: Theorems of Mordell and Tijdeman Figure: Louis Mordell (left) and Robert Tijdeman (right)

15 Gaussian Integers A complex number is a number of the form a + bi, where a and b are real numbers and i satisfies the equation i = 0. A number a + bi with a,b rational integers is called a Gaussian integer. The set of all Gaussian integers is denoted by Z[i]. Exercise. Let a + bi,c + di be Gaussian integers. Prove the following: 1. Every rational integer is a Gaussian integer; 2. (a + bi) + (c + di) is a Gaussian integer; 3. (a + bi) (c + di) is a Gaussian integer; 4. (a + bi)(c + di) is a Gaussian integer. Sets where we can add, subtract and multiply are called rings. More formally, A when for α,β A we have α ± β A and αβ A.

16 Divisibility and Norm Let a,b be Gaussian integers. We say that a divides b when b = ak for some Gaussian integer k. We write a b in this case, and a b otherwise. The value a 2 + b 2 is called the norm of a Gaussian integer a + bi. It is denoted by N(a + bi). Exercise. Prove that 1 + 2i divides 5 and does not divide 7. Exercise. Let α, β be Gaussian integers. Prove that N(αβ) = N(α)N(β). Therefore the norm function is multiplicative. Exercise. Prove that N(α) 0 for all Gaussian integers α and N(α) = 0 if and only if α = 0.

17 Units and Primes In a ring A there may exist special numbers that divide 1. Such elements are called units. For example, the only in units in Z are 1 and 1. Exercise. Show that if α is a Gaussian unit then N(α) = 1. Exercise. Prove that the units of Z[i] are 1, 1,i and i. A Gaussian integer α is called a Gaussian prime if it is not a unit and any factorization α = βγ in Z[i] forces β or γ to be a unit. Exercise. Find Gaussian primes among the integers 2, 3, 5, 7. Just like rational primes, Gaussian primes have the following property: if γ is a Gaussian prime and γ αβ, then either γ α or γ β. Remember this property: you will need in the next exercise!

18 The Remainder Theorem, GCD and the Fundamental Theorem of Arithmetic The Remainder Theorem. Let a, b be rational integers, a > 0. Then there exist unique integers q and r such that b = aq + r, where 0 r < a. The Remainder Theorem for Gaussian Integers. Let a, b be Gaussian integers. Then there exist Gaussian integers q and r such that b = aq + r, where N(r) < N(a). Let a and b be integers. An integer g such that g a and g b, with N(g) the largest, is called the greatest common divisor of a and b. It is denoted by gcd(a,b). The Fundamental Theorem of Arithmetic. Up to multiplication by a unit, any non-zero Gaussian integer can be written uniquely (up to reordering) as the product of Gaussian primes.

19 THE SUM OF SQUARES

20 The Sum of Squares In this exercise we will investigate which numbers n can be written as the sum of two squares. That is, n = a 2 + b 2 for some integers a and b. Exercise. Compute first 10 numbers that are sums of two squares. Step 1. Let m and n be positive integers that are sums of two squares. Prove that mn is also a sum of two squares. Hint: use the fact that the norm N is multiplicative. Step 2. Prove that every integer that is a sum of two squares is 0,1 (mod 4). Conclude that every rational prime p 3 (mod 4) is not a sum of two squares, and so it is a Gaussian prime.

21 The Sum of Squares Step 3. Let p be a rational prime such that p 1 (mod 4). In this exercise, we will use the fact that there always exists an integer x such that p x Show that p does not divide neither x + i nor x i. Conclude that it is not prime, so p = αβ for some Gaussian integers α,β. 2. Prove that neither α nor β are units. Conclude that N(α) = p, so p is a sum of two squares. Step 4. Show that 2 is a sum of 2 squares. Conclude that every number of the form 2 t p e pe k k q2f q2f l l is a sum of two squares, where p i are primes that are 1 (mod 4) and q i are primes that are 3 (mod 4).

22 Next Time We will see why most of this theory fails for other rings, such as Z[ 5]. Learn more about algebraic numbers!

23 THANK YOU FOR COMING!

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