Introduction to Algebraic Number Theory Part I · Number Theories I Number theory studies properties of numbers, such as 2; 1;22=7, p 2, or p. I There are many subareas of number

Introduction to Algebraic Number TheoryPart I

A. S. Mosunov

University of WaterlooMath Circles

November 7th, 2018

Goals

I Explore the area of mathematics called Algebraic NumberTheory.

I Specifically, we will see how to generalize the notions ofintegers, rational numbers, prime numbers, etc.

I Goal 1. Understand the basics of the theory.

I Goal 2. See beautiful theorems.

I Goal 3. Understand open problems.

Number Theories

I Number theory studies properties of numbers, such as2,−1,22/7,

√2, or π.

I There are many subareas of number theory, such as Analyticnumber theory, Theory of Diophantine approximation, etc.

I Algebraic number theory studies numbers that are roots ofpolynomial equations, such as

−3, which is a root of x + 3 = 0,√2, which is a root of x2−2 = 0,i , which is a root of x2 + 1 = 0.

I Transcendental number theory studies numbers that do not

satisfy this property, such as π, log 2 or√

2√2.

I Determining whether a number is algebraic or transcendental

can be very hard! Is√

2√2

+ π transcendental?

Why Study Number Theory?

Figure: Messaging apps that (hopefully!) use cryptographic protocolsbased on hard number theoretical problems

Why Study Number Theory?

I It is beautiful.

I It is applicable! Many cryptographic protocols reside ondifficult number theoretical problems.

I Many protocols, such as RSA or the Diffie-Hellman Protocol,which are based on “regular” number theory are vulnerable toquantum computer attacks.

I Algebraic number theory comes to the rescue!

I Lattice-based cryptography is quantum-safe and it usesproperties of numbers that are roots of xn + 1 = 0.

I CSIDH is a cryptographic protocol that is quantum-safe and ituses properties of numbers of the form a+b

√m, where m is a

very small negative integer and a,b are rational numbers.

BACKGROUND

Rational Integers

I The numbers . . . ,−2,−1,0,1,2, . . . are called (rational)integers. The set of all integers is denoted by Z.

I Let a and b be integers. We say that a divides b when b = akfor some integer k . We write a | b in this case, and a - botherwise.

I A number p ≥ 2 is a (rational) prime if it is divisible only by 1and p.

I The Fundamental Theorem of Arithmetic. Any integergreater than 1 can be written uniquely (up to reordering) asthe product of primes.

I Let a and b be integers. The largest integer g such that g | aand g | b is called the greatest common divisor of a and b.It is denoted by gcd(a,b).

I The numbers a and b are called coprime if gcd(a,b) = 1.

Detour: Rational Numbers and Apery’s TheoremI 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 rationalnumbers is denoted by Q.

I Determining whether a given number is rational or irrationalcan be very hard!

I In 1979 the French mathematician Roger Apery proved thatthe number

ζ (3) = 1 +1

23+

1

33+

1

43+

1

53+ . . .≈ 1.2020569031 . . .

is irrational.I 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 WadimZudilin in the 90’s).

I See the article A proof that Euler missed by Alfred van derPoorten:https://web.archive.org/web/20110706114957/http:

//www.maths.mq.edu.au/~alf/45.pdf.

Detour: Rational Numbers and Apery’s Theorem

Figure: Roger Apery (1916 – 1994)

Exercise

I If a,b are coprime positive integers and ab = c2 for someinteger c, show that a = t2 and b = s2 for some integers t ands.

I Show that for any integer x the numbers x and x2 + 1 arecoprime.

I Numbers 0,1,22 = 4,32 = 9, . . . are called squares. Show thatthe distance between k2 and (k + 1)2 is equal to 2k + 1.When is this distance equal to 1?

I Use the previous results to conclude that the equationy2 = x3 + x has no solutions in positive integers x and y .

ALGEBRAIC NUMBER THEORY

BEGINS

How Euler “Almost” Discovered Algebraic NT

I Can the distance between a square and a cube be equal toone?

I In 1700’s, Euler showed that the only square and cube thatdiffer by 1 are 8 and 9.

I Homework. Prove that the equation y2 = x3 + 1 has onlyone solution in positive integers. Hint: use the fact that

(x3 + 1) = (x + 1)(x2−x + 1).

I He also “almost” proved that the only square and cube thatdiffer by 2 are 25 and 27.

I Idea: consider the equation y2 = x3−2 and write it as

(y +√−2)(y −

√−2) = x3.

If y +√−2 and y −

√−2 are “coprime”, they must be

“cubes”. But what does “coprime” even mean in this setting?

Detour: Theorems of Mordell and Tijdeman

I We have already seen that the distance between consecutivesquares grows. Same observation applies to cubes.

I Does the distance between consecutive squares and cubesgrow?

0,1,4,8,9,16,25,27,36,49,64,81,100,121,125,144, . . .

I The answer is yes. This was proved by the Britishmathematician Loius Mordell in 1960’s.

I In 1976, Robert Tijdeman showed that the number ofconsecutive powers that differ by 1 is finite. Questions aboutlarger distances is still open.

I The solutions (x ,y ,m,n) to the equation ym = xn + 1 mustsatisfy

|x |, |y |,m,n ≤ eeee

730

.

Detour: Theorems of Mordell and Tijdeman

Figure: Louis Mordell (left) and Robert Tijdeman (right)

Gaussian Integers

I A complex number is a number of the form a+bi , where aand b are real numbers and i satisfies the equation i2 + 1 = 0.

I A number a+bi with a,b rational integers is called aGaussian integer. The set of all Gaussian integers is denotedby Z[i ].

I Exercise. Let a+bi ,c +di be Gaussian integers. Prove thefollowing:

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.

I Sets where we can add, subtract and multiply are called rings.More formally, A when for α,β ∈ A we have α±β ∈ A andαβ ∈ A.

Divisibility and Norm

I Let a,b be Gaussian integers. We say that a divides b whenb = ak for some Gaussian integer k . We write a | b in thiscase, and a - b otherwise.

I The value a2 +b2 is called the norm of a Gaussian integera+bi . It is denoted by N(a+bi).

I Exercise. Prove that 1 + 2i divides 5 and does not divide 7.

I Exercise. Let α,β be Gaussian integers. Prove that

N(αβ ) = N(α)N(β ).

Therefore the norm function is multiplicative.

I Exercise. Prove that N(α)≥ 0 for all Gaussian integers α

and N(α) = 0 if and only if α = 0.

Units and Primes

I In a ring A there may exist special numbers that divide 1.Such elements are called units. For example, the only in unitsin Z are −1 and 1.

I Exercise. Show that if α is a Gaussian unit then N(α) = 1.

I Exercise. Prove that the units of Z[i ] are 1,−1, i and −i .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 aunit.

I Exercise. Find Gaussian primes among the integers 2,3,5,7.

I Just like rational primes, Gaussian primes have the followingproperty: if γ is a Gaussian prime and γ | αβ , then either γ | αor γ | β . Remember this property: you will need in the nextexercise!

The Remainder Theorem, GCD and the FundamentalTheorem of Arithmetic

I The Remainder Theorem. Let a,b be rational integers,a> 0. Then there exist unique integers q and r such thatb = aq+ r , where 0≤ r < a.

I The Remainder Theorem for Gaussian Integers. Let a,bbe Gaussian integers. Then there exist Gaussian integers qand r such that b = aq+ r , where N(r) < N(a).

I Let a and b be integers. An integer g such that g | a andg | b, with N(g) the largest, is called the greatest commondivisor of a and b. It is denoted by gcd(a,b).

I The Fundamental Theorem of Arithmetic. Up tomultiplication by a unit, any non-zero Gaussian integer can bewritten uniquely (up to reordering) as the product of Gaussianprimes.

THE SUM OF SQUARES

The Sum of Squares

In this exercise we will investigate which numbers n can be writtenas the sum of two squares. That is, n = a2 +b2 for some integers aand 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 twosquares. Prove that mn is also a sum of two squares. Hint: usethe fact that the norm N is multiplicative.

Step 2. Prove that every integer that is a sum of two squares is ofthe form 4k, 4k + 1 or 4k + 2 for some integer k . Conclude thatevery rational prime p of the form 4k + 3 is not a sum of twosquares, and so it is a Gaussian prime.

The Sum of Squares

Step 3. Let p be a rational prime of the form 4k + 1 for someinteger k . In this exercise, we will use the fact that there alwaysexists an integer x such that p | x2 + 1.

1. Show that p does not divide neither x + i nor x− i . Concludethat it is not prime, so p = αβ for some Gaussian integersα,β neither of which is a unit.

2. Prove 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 everynumber of the form

2tpe11 . . .pekk q2f11 . . .q2f``

is a sum of two squares, where pi are primes that are of the form4k + 1 and qi are primes of the form 4k + 3.

Next Time

I We will see why most of this theory fails for other rings, suchas Z[

√−5].

I Learn more about algebraic numbers!

THANK YOU FOR COMING!