algebra and number theory a selection of highlights pdf

1 The natural, integral and rational numbers1.1 Number theory and axiomatic systems Number theory begins as the study of the whole numbers or counting numbers.. For-mally the counting nu

Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger,Dennis Spellman

Algebra and Number Theory

De Gruyter Textbook

Geometry and Discrete Mathematics A Selection of Highlights

Benjamin Fine, Anthony Gaglione, Anja Moldenhauer,

Gerhard Rosenberger, Dennis Spellman, 2018

ISBN 978-3-11-052145-0, e-ISBN (PDF) 978-3-11-052150-4,

e-ISBN (EPUB) 978-3-11-052153-5

Discrete Algebraic Methods.

Arithmetic, Cryptography, Automata and Groups

Volker Diekert, Manfred Kufleitner, Gerhard Rosenberger,

Ulrich Hertrampf, 2016

ISBN 978-3-11-041332-8, e-ISBN (PDF) 978-3-11-041333-5,

e-ISBN (EPUB) 978-3-11-041632-9

A Course in Mathematical Cryptography

Gilbert Baumslag, Benjamin Fine, Martin Kreuzer,

The Elementary Theory of Groups.

A Guide through the Proofs of the Tarski Conjectures

Benjamin Fine, Anthony Gaglione, Alexei Myasnikov,

Gerhard Rosenberger, Dennis Spellman, 2014

ISBN 978-3-11-034199-7, e-ISBN (PDF) 978-3-11-034203-1,

e-ISBN (EPUB) 978-3-11-038257-0

Abstract Algebra.

Applications to Galois Theory, Algebraic Geometry and Cryptography

Celine Carstensen, Benjamin Fine, Gerhard Rosenberger, 2011ISBN 978-3-11-025008-4, e-ISBN (PDF) 978-3-11-025009-1

Benjamin Fine, Anthony Gaglione,

Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman

Algebra and

Number Theory

|

A Selection of Highlights

Prof Dr Anthony Gaglione

United States Naval Academy

20146 Hamburg Germany Prof Dr Dennis Spellman Temple University Department of Mathematics

1801 N Broad Street Philadelphia, PA 19122 USA

ISBN 978-3-11-051584-8

e-ISBN (PDF) 978-3-11-051614-2

e-ISBN (EPUB) 978-3-11-051626-5

Library of Congress Cataloging-in-Publication Data

A CIP catalog record for this book has been applied for at the Library of Congress.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Typesetting: VTeX UAB, Lithuania

Printing and binding: CPI books GmbH, Leck

Cover image: agsandrew / iStock / Getty Images Plus

♾ Printed on acid-free paper

Printed in Germany

www.degruyter.com

To many students, as well as to many teachers, mathematics seems like a mundanediscipline, filled with rules and algorithms and devoid of beauty and art However tosomeone who truly digs deeply into mathematics this is quite far from the truth Theworld of mathematics is populated with true gems; results that both astound and point

to a unity in both the world and a seemingly chaotic subject It is often that these gemsand their surprising results are used to point to the existence of a force governing the

universe; that is, they point to a higher power Euler’s magic formula, e iπ+1 = 0, which

we go over and prove in this book is often cited as a proof of the existence of God While

to someone seeing this statement for the first time it might seem outlandish, however ifone delves into how this result is generated naturally from such a disparate collection

of numbers it does not seem so strange to attribute to it a certain mystical significance.Unfortunately most students of mathematics only see bits and pieces of this amaz-ing discipline In this book, which we call Algebra and Number Theory, we intro-duce and examine many of these exciting results We planned this book to be used incourses for teachers and for the general mathematically interested so it is somewhatbetween a textbook and just a collection of results We examine these mathematicalgems and also their proofs, developing whatever mathematical results and techniques

we need along the way In Germany and the United States we see the book as a MastersLevel Book for prospective teachers

With the increasing demand for education in the STEM subjects, there is the ization that to get better teaching in mathematics, the prospective teachers must both

real-be more knowledgeable in mathematics and excited about the subject The courses inteacher preparation do not touch many of these results that make the discipline so ex-citing This book is intended to address this issue The first volume is on Algebra andNumber Theory We touch on numbers and number systems, polynomials and poly-nomial equations, geometry and geometric constructions These parts are somewhatindependent so a professor can pick and choose the areas to concentrate on Muchmore material is included than can be covered in a single course We prove all rele-vant results that are not too technical or complicated to scare the students We findthat mathematics is also tied to its history so we include many historical comments

We try to introduce all that is necessary however we do presuppose certain jects from school and undergraduate mathematics These include basic knowledge

sub-in algebra, geometry and calculus as well as some knowledge of matrices and lsub-inearequations Beyond these the book is self-contained

This first volume of two is called Algebra and Number Theory There are fourteenchapters and we think we have introduced a very wide collection of results of the typethat we have alluded to above In Chapters 1–5 we look at highlights on the integers Weexamine unique factorization and modular arithmetic and related ideas We show howthese become critical components of modern cryptography especially public key cryp-

tographic methods such as RSA Three of the authors (Fine, Moldenhauer and berger) work partly as cryptographers so cryptography is mentioned and explained

Rosen-in several places In Chapters 4 and 5 we look at exceptional classes of Rosen-integers such

as the Fibonacci numbers as well as the Fermat numbers, Mersenne numbers, perfectnumbers and Pythagorean triples We explain the golden section as well as expressingintegers as sums of squares In Chapters 6–8 we look at results involving polynomi-als and polynomial equations We explain field extensions at an understandable leveland then prove the insolvability of the quintic and beyond The insolvability of thequintic in general is one of the important results of modern mathematics

In Chapters 9–12 we look at highlights from the real and complex numbers leadingeventually to an explanation and proof of the Fundamental Theorem of Algebra Along

the way we consider the amazing properties of the numbers e and π and prove in detail

that these two numbers are transcendent

Chapter 13 is concerned with the classical problem of geometric constructions anduses the material we developed on field extensions to prove the impossibility of certainconstructions

Finally in Chapter 14 we look at Euclidean Vector Spaces We give several ric applications and look for instance at a secret sharing protocol using the closestvector theorem

geomet-We would like to thank the people who were involved in the preparation of themanuscript Their dedicated participation in translating and proofreading are grate-fully acknowledged In particular, we have to mention Anja Rosenberger, AnnikaSchürenberg and the many students who have taken the respective courses in Dort-mund, Fairfield and Hamburg Those mathematical, stylistic, and orthographic errorsthat undoubtedly remain shall be charged to the authors Last but not least, we thank

de Gruyter for publishing our book

Benjamin FineAnthony GaglioneAnja MoldenhauerGerhard RosenbergerDennis Spellman

Preface|V

1 The natural, integral and rational numbers|1

1.1 Number theory and axiomatic systems|1

1.2 The natural numbers and induction|1

1.3 The integers ℤ|10

1.4 The rational numbers ℚ|13

1.5 The absolute value in ℕ, ℤ and ℚ|15

2 Division and factorization in the integers|19

2.1 The Fundamental Theorem of Arithmetic|19

2.2 The division algorithm and the greatest common divisor|23

2.3 The Euclidean algorithm|26

2.4 Least common multiples|30

2.5 General gcd’s and lcm’s|33

3 Modular arithmetic|39

3.1 The ring of integers modulo n|39

3.2 Units and the Euler φ-function|43

3.3 RSA cryptosystem|46

3.4 The Chinese Remainder Theorem|47

3.5 Quadratic residues|54

4 Exceptional numbers|61

4.1 The Fibonacci numbers|61

4.1.1 The golden rectangle|67

4.1.2 Squares in semicircles|68

4.1.3 Side length of a regular 10-gon|69

4.1.4 Construction of the golden section α with compass and straightedge

from a given a ∈ ℝ, a > 0|70

4.2 Perfect numbers and Mersenne numbers|71

5 Pythagorean triples and sums of squares|83

5.1 The Pythagorean Theorem|83

5.2 Classification of the Pythagorean triples|85

5.3 Sum of squares|89

6 Polynomials and unique factorization|95

6.1 Polynomials over a ring|95

6.2 Divisibility in rings|98

6.3 The ring of polynomials over a field K|100

6.3.1 The division algorithm for polynomials|101

6.3.2 Zeros of polynomials|103

6.5 The Euclidean algorithm and greatest common divisor of polynomials

over fields|112

6.5.1 The Euclidean algorithm for K[x]|114

6.5.2 Unique factorization of polynomials in K[x]|115

6.5.3 General unique factorization domains|116

6.6 Polynomial interpolation and the Shamir secret sharing scheme|117

6.6.1 Secret sharing|117

6.6.2 Polynomial interpolation over a field K|117

6.6.3 The Shamir secret sharing scheme|121

7 Field extensions and splitting fields|125

7.1 Fields, subfield and characteristic|125

9.1 The real number system|157

9.2 Decimal representation of real numbers|168

9.3 Periodic decimal numbers and the rational number|172

9.4 The uncountability of ℝ|173

9.5 Continued fraction representation of real numbers|175

9.6 Theorem of Dirichlet and Cauchy’s Inequality|176

9.7.1 Normed fields and Cauchy completions|179

9.7.2 The p-adic fields|180

9.7.3 The p-adic norm|183

10.1 The field ℂ of complex numbers|189

10.2 The complex plane|193

10.2.1 Geometric interpretation of complex operations|196

10.2.2 Polar form and Euler’s identity|197

10.2.3 Other constructions of ℂ|201

10.2.4 The Gaussian integers|201

10.3 The Fundamental Theorem of Algebra|202

10.3.1 First proof of the Fundamental Theorem of Algebra|204

10.3.2 Second proof of the Fundamental Theorem of Algebra|207

10.4 Solving polynomial equations in terms of radicals|209

10.5 Skew field extensions of ℂ and Frobenius’s Theorem|220

11 Quadratic number fields and Pell’s equation|227

11.1 Algebraic extensions of ℚ|227

11.2 Algebraic and transcendental numbers|228

11.3 Discriminant and norm|230

11.4 Algebraic integers|235

11.4.1 The ring of algebraic integers|236

11.5 Integral bases|238

11.6 Quadratic fields and quadratic integers|240

12 Transcendental numbers and the numbers e and π|249

12.1 The numbers e and π|249

12.1.1 Calculation e of π|251

12.2 The irrationality of e and π|256

12.3.1 The normal distribution|263

12.3.2 The Gamma Function and Stirling’s approximation|264

12.3.3 The Wallis Product Formula|266

12.4 Existence of a transcendental number|270

12.5 The transcendence of e and π|273

12.6 An amazing property of π and a connection to prime numbers|282

13 Compass and straightedge constructions and the classical

problems|289

13.1 Historical remarks|289

13.2 Geometric constructions|289

13.3 Four classical construction problems|296

13.3.1 Squaring the circle (problem of Anaxagoras 500–428 BC)|296

13.3.2 The doubling of the cube or the problem from Deli|296

13.3.3 The trisection of an angle|297

13.3.4 Construction of a regular n-gon|298

14 Euclidean vector spaces|303

14.1 Length and angle|303

14.2 Orthogonality and Applications in ℝ2and ℝ3 |309

14.3 Orthonormalization and closest vector|317

14.4 Polynomial approximation|321

14.5 Secret sharing scheme using the closest vector theorem|323 Bibliography|327

Index|329

1 The natural, integral and rational numbers

1.1 Number theory and axiomatic systems

Number theory begins as the study of the whole numbers or counting numbers

For-mally the counting numbers 1,2,… are called the natural numbers and denoted by ℕ.

If we add to this the number zero, denoted by 0, and the negative whole numbers we

get a more comprehensive system called the integers which we denote by ℤ The focus

of this book is on important and sometimes surprising results in number theory andthen further results in algebra Many results in number theory, as we shall see, seemlike magic In order to rigorously prove these results we place the whole theory in anaxiomatic setting which we now explain

In mathematics, when developing a concept or a theory it is often not possible, allused terms, properties or claims to prove, especially existence of some mathematicalfundamentals One can solve this problem then by an axiomatic approach The basis

of a theory then is a system of axioms:

– Certain objects and certain properties of these objects are taken as given and cepted

ac-– A selection of statements (the axioms) are considered by definition as true and

evident

A theorem in the theory then is a true statement, whose truth can be proved from the

axioms with help of true implications A system of axioms is consistent if one can not

prove a statement of the form “A and not A” The verification is in individual cases

often a complicated or even an unsolvable problem We are satisfied, if we can quote

a model for the system of axioms, that is, a system of concrete objects, which meet all the given axioms A system of axioms is called categorical if essentially there exists

only one model By this we mean that for any two models we always get from one

model to the other by renaming of the objects If this is true then we have an axiomatic

characterization of the model.

In the next section we introduce the natural numbers axiomatically

1.2 The natural numbers and induction

The natural numbers ℕ are presented by the system of axioms developed by G Peano(1858–1932) This is done as follows

The set ℕ of the natural numbers is described by the following axioms:

(ℕ 1) 1 ∈ ℕ

(ℕ 2) Each a ∈ ℕ has exactly one successor a+∈ ℕ

(ℕ 3) Always is a+≠1, and for each b ≠ 1 there exists an a ∈ ℕ with b = a+

(ℕ 4) a ≠ b ⇒ a+≠b+

(ℕ 5) If T ⊂ ℕ, 1 ∈ T, and if together with a ∈ T also a+∈T, then T = ℕ.

(Axiom of mathematical induction or just induction.)

Remarks 1.1 (1) (ℕ 2) and (ℕ 4) mean that the map

(iii) a < b ∶⇔ ∃ x ∈ ℕ with a + x = b (“a smaller than b”),

a ≤ b ∶⇔ a = b or a < b (“a equal or smaller than b”).

We need to recall some definitions

A semigroup is a set H ≠ ∅ together with a binary operation ⋅ ∶ H × H → H that satisfies the associative property for all a,b,c ∈ H:

multi-A monoid S is a semigroup with a unity element e, that is, an element e with a ⋅ e =

a = e ⋅ a for all a ∈ S; e is uniquely determined.

Moreover, a monoid S is called a group if for each a ∈ S there exists an inverse element a−1∈S with aa−1=a−1a = e The monoid or group is named commutative or

abelian if in addition

a ⋅ b = b ⋅ a for all a,b ∈ S.

We often write 1 instead of e We also often drop ⋅ and use just juxtaposition for this operation If we use the addition + we often write 0 instead of e and call 0 the zero

element of S.

Theorem 1.2 (1) The addition for ℕ is associative, that is,

a + (b + c) = (a + b) + c,

1.2 The natural numbers and induction | 3

and commutative, that is,

a + b = b + a.

This means, ℕ is a commutative semigroup with respect to the addition.

(2) The multiplication for ℕ is associative, that is,

a(bc) = (ab)c, and commutative, that is,

(5) If a ≤ b and c ≤ d then a + c ≤ b + d and ac ≤ bd.

Proof The statements follow directly from the definition and the Peano axioms We

leave the proofs as an exercise As an example we prove (3) using (1) and (2): Let

a,b ∈ ℕ be arbitrary and T ⊂ ℕ the set of the c ∈ ℕ with (a + b)c = ac + bc We have

As usual we write a n for a ⋅ a⋯a⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

n times and na for a + a + ⋯ + a⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

n times , when a,n ∈ ℕ.

Remarks 1.3 (1) By the development of the addition in ℕ we suggest the usual

rep-resentation of natural numbers as numerals:

(3) Theorem 1.2 also allows to subtract smaller natural numbers from larger ones If

a,b ∈ ℕ with a < b, then there is an x ∈ ℕ with a + x = b We define the subtraction

(4) The mathematical proof technique mathematical induction is based on the Peano

axiom (ℕ 5) It is a form of direct proof, and it is done in two steps

The first step, known as the base case, is to prove the given statement A(n), which

is definable for all n ∈ ℕ, for the first natural number 1 The second step, known as the induction step, is to prove that the given statement A(n) is true for any natural number n implies the given statement is true for the next natural number In other words, if A(1) is true and if we can show that under the assumption that A(n) is true for any n, then A(n + 1) is true, then A(n) is true for all n ∈ ℕ.

We call the mathematical induction the first induction principle or the principle of

mathematical induction (PMI).

It is clear that we may start with the mathematical induction with any natural

number n0>1 instead of 1, we just need a base This can be done with the

Proof Let A(n), n ∈ ℕ, be the asserted statement.

(a) A(1) is true because

1.2 The natural numbers and induction | 5

We remark that the n-th triangular number T n , n ∈ ℕ, is defined as n(n+1)2

Geomet-rically T n is the number of dots composing a regular triangle with n dots on a side,

see Figure 1.1

Figure 1.1: Geometrical representation of the triangular numbers T1, T2, T3and T4.

From this geometrical representation (see Figure 1.1) in a regular triangle we see

by completing the triangle to a square that

The children had to calculate the sum of all numbers from 1 to 100 Gauss realizedthe pattern:

n

∑

k=1 k3=n2(n + 1)2

4 =T n2,for n ∈ ℕ.

There is a beautiful general formula by Al-Haitam (965–1038):

(n + 1)∑n

i=1 i k=∑n i=1 i k+1+∑n p=1

(

p

∑

i=1 i k)

We present a geometrical proof, see Figure 1.2, which is still remarkable

Figure 1.2: Geometrical proof of the formula

by Al-Haitam.

Recall that 1k+1=1k, 2 ⋅ 2k=2k+1, 3 ⋅ 3k=3k+1, …

(2) Claim.

2n>n2 for all n ≥ 5.

Proof Let A(n), n ≥ 5, be the asserted statement.

(a) A(5) is true because

(3) A polygon in the plane ℝ2with n+2 sides, n ∈ ℕ, is called convex, if the connecting

line segment between any two points of the polygon is totally within the polygon

Examples 1.6 We give examples for convex polygons in Figure 1.3.

1.2 The natural numbers and induction | 7

Figure 1.3: Convex polygons.

Claim The angle sum W n of the interior angles in a convex polygon with n+2, n ∈ ℕ, sides is n ⋅ 180∘.

Proof (a) If n = 1 we have a triangle from which it is known that W1=180∘

(b) Assume that the assert statement holds for n ∈ ℕ We consider a convex gon with (n + 1) + 2 sides We divide the polygon into a triangle and a polygon with

q − 1 +q n+1=q

n+2−q

q − 1 .

We remark that

1 +∑n

i=1 q i=q n+1−1

q − 1 .

(5) Claim Let M be an arbitrary set with n (distinct) elements, n ∈ ℕ Let 𝒫(M) be the

power set of M Then |𝒫(M)| = 2 n for the number of elements in 𝒫(M).

Proof (a) If n = 1 then M = {a} for some a, and hence, 𝒫(M) = {∅,{a}}, that is,

|𝒫(M)| = 2 = 21

(b) Assume that the asserted statement holds for each set with n elements Let,

without loss of generality,

M = {a1,a2, … ,a n+1} = {a1,a2, … ,a n} ∪ {a n+1} =M′∪ {a n+1}

be a set with n + 1 elements By the induction hypothesis

|𝒫(M′)| =2n.Now, let A ⊂ M be a subset of M Then we have exactly one of the following cases:

ordering property This is the following Let S be a nonempty subset of the natural bers ℕ Then S has a least element.

num-We will abbreviate the least well-ordering property by LWO

In the next theorem below we show that the principle of mathematical induction(PMI) is equivalent to the LWO By equivalent, we mean here that if we assume thatthe PMI is true then we can prove the LWO and if we assume the LWO is true then wecan prove the PMI

Theorem 1.7 The principle of mathematical induction is equivalent to the least

well-ordering property.

Proof To prove this we must assume first the principle of mathematical induction and

show that the well-ordering property holds and then vice versa Suppose that the PMI

holds and let S be a nonempty subset of ℕ We must show that S has a least element.

We let T be the set

T = {x ∈ ℕ ∣ x ≤ s, for all s ∈ S}.

1.2 The natural numbers and induction | 9

Now 1 ∈ T since S is a subset of ℕ If whenever x ∈ T it were to follow that (x + 1) ∈ T, then by the inductive property T = ℕ but then S would be empty contradicting that S

is nonempty Therefore there exists an a with a ∈ T and (a + 1) ∉ T We claim that a is the least element of S Now a ≤ s for all s ∈ S because a ∈ T If a ∉ S then every s ∈ S would also satisfy (a+1) ≤ s This would imply that (a+1) ∈ T which is a contradiction Therefore a ∈ S and a ≤ s for all s ∈ S and hence a is the least element of S.

Conversely suppose the well-ordering property holds and suppose that S is a set of ℕ with the properties that 1 ∈ S and that whenever n ∈ S it follows that (n+1) ∈ S.

sub-We must show that S = ℕ If S ≠ ℕ, then the set difference ℕ⧵S, that is, the set of all ements in ℕ but not in S, would be a nonempty subset of ℕ Thus by the LWO, it has a least element, say n Hence (n − 1) is not in ℕ ⧵ S or (n − 1) ∈ S But then by the assumed property of S we get that (n − 1) + 1 = n ∈ S which gives a contradiction Therefore ℕ ⧵ S

el-is empty and S = ℕ.

Based on Theorem 1.7 we have a second form of mathematical induction that we

call the second induction principle This is also known as course of values induction or

strong induction

Theorem 1.8 Let A(n) be a statement which is defined for all n ∈ ℕ If A(1) is true and

if we can show that under the assumption that A(k) is true for all k ∈ ℕ with k < n for any n ∈ ℕ, then A(n) is true for all n ∈ ℕ.

Proof Let T = {n ∈ ℕ ∣ A(n) is not true} ⊂ ℕ Assume that T ≠ ∅ Then by LWO T

con-tains a smallest element, which means that there is an n ∈ ℕ, n > 1, with A(n) not true but A(1) is true and A(k) is true for all k ∈ ℕ, k < n But this contradicts our hypothesis Therefore A(n) is true for all n ∈ ℕ.

Corollary 1.9 The two principles for mathematical induction are equivalent.

Proof If the second principle holds then certainly also the first one If the first

princi-ple holds then also the second one vie Theorem 1.8

Remarks 1.10 (1) Also for the second induction principle we may take as the base

Claim For n ≥ 2 the Fibonacci number f n is the number of all 0–1-sequences of length

n − 2 which do not contain neighboring 1’s.

Proof Let M n be the set of all 0–1-sequences of length n−2, n ≥ 2, which do not contain neighboring 1’s We have |M2| =f2=1, the empty sequence, and |M3| =f3=2 because

we just have the sequences 0 and 1

Now, let n ≥ 4, and we assume that the statement is true for all k with 2 ≤ k < n Let

M n=M n(0)∪M n(1)be the disjoint union of M n(0), the set of the sequences in M nending

with a 0, and M n(1), the set of the sequences in M nending with a 1 Because of our

condition, in fact, each sequence in M n(1)has to end with 01 (recall that we do not haveneighboring 1’s) Therefore

|M n| = |M n(0)| + |M n(1)| = |M n−1| + |M n−2| =f n−1+f n−2=f n.

Remark 1.12 We may establish the natural numbers constructive from the axiomatic

set theory One starts with the axiomatically existent empty set ∅ and gets for each set

X a successor set using the axiom that there exists an infinite set, that is, there is a set

which contains with ∅ and X also the set X ∪ {x} One then defines

x = a − b for a,b ∈ ℕ and a > b.

This means algebraically that we may solve the equation

1.3 The integers ℤ | 11

Historically it took a long time before the negative numbers were accepted In cient times and in parts of the middle ages mathematicians like Al Khwarizmi (ca.780–850), Cardano (1501–1576) and Vieta (1540–1603) considered the negative num-bers as forbidden or worked with them solely symbolically believing that these nega-tive numbers do not exist This can be seen from the many quadratic equations over

an-ℕwhich in their feeling are forbidden or not-existent

We consider the set

ℕ × ℕ = {(a,b) ∣ a,b ∈ ℕ},

the set of pairs of natural numbers

If a > b and c > d, then we may have the equation a−b = c−d in ℕ, or equivalently,

a + d = c + b This is the inspiring background for introducing the integers We define

an addition and multiplication on ℕ × ℕ as follows:

(a,b) + (c,d) = (a + c,b + d),

(a,b) ⋅ (c,d) = (ac + bd,ad + bc).

With respect to the above equation a + d = b + c whenever a > b,c > d and a − b = c − d,

we introduce a relation on ℕ × ℕ:

(a,b) ∼ (c,d) ⇔ a + d = c + b.

This is an equivalence relation Certainly, (a,b) ∼ (a,b) and (c,d) ∼ (a,b) if (a,b) ∼

(c,d), that is, the relation is reflexive and symmetric It is also transitive, that is, if

(a,b) ∼ (c,d) and (c,d) ∼ (e,f ) then (a,b) ∼ (e,f ) because from a + b = c + d and c + d =

(a,b) ⋅ (c,d) ∼ (a′,b′) ⋅ (c′,d′)

We leave the proof for this as an exercise Together with these operations ℤ becomes

a commutative ring with unity

Remark 1.13 We remind that a ring is a set R ≠ ∅ equipped with two binary

opera-tions + ∶ R × R → R and ⋅ ∶ R × R → R satisfying the following three sets of properties for all a,b,c ∈ R:

– R is a commutative group under addition, that is,

(1) (a + b) + c = a + (b + c).

(2) a + b = b + a.

(3) There is a zero element 0 ∈ R such that a + 0 = a for all a ∈ R.

(4) For each a ∈ R exists −a ∈ R such that a + (−a) = 0.

We call −a the negative element of a.

– R is a semigroup under multiplication, that is,

(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

– The multiplication is distributive with respect to the addition, that is,

a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) and

(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a).

A commutative ring with unity 1 is a ring R, in which the semigroup under

multiplica-tion is also commutative and has the unity element 1

In the case ℤ the zero element is the class represented by (1,1), and the unityelement is the class represented by (2,1)

We briefly write (a,b) for the equivalence class represented by (a,b) The alence class (a,b) has for a > b a unique representative of the form (n + 1,1), where

equiv-n = a − b, aequiv-nd for b > a a uequiv-nique represeequiv-ntative of the form (1,m + 1), where m = b − a.

This gives a possibility to embed ℕ into ℤ We define the map

φ ∶ ℕ → ℤ

n ↦ (n + 1,1).

The map φ is injective and satisfies

φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b)

for a,b ∈ ℕ This means, that φ is an embedding.

Therefore we may identify ℕ with φ(ℕ) ⊂ ℤ, and we write after the identification

n = (n + 1,1) for n ∈ ℕ.

We write −n for the equivalence class (1,n + 1) This is reasonable because

(n + 1,1) + (1,n + 1) = (1,1),

the zero element of ℤ Therefore we now write 0 for the class (1,1)

In this sense, ℤ is the disjoint union

ℤ = ℕ−∪ {0} ∪ ℕ,where ℕ−= {−n ∣ n ∈ ℕ}.

With this, the addition and multiplication in ℤ is just according to customs

1.4 The rational numbers ℚ | 13

We now transfer the ordering in ℕ to an ordering in ℤ We define a < b in ℤ if there exists an n ∈ ℕ with a + n = b, and a ≤ b if a = b or a < b This ordering is compatible

with the addition and the multiplication in ℤ, and we get directly from the definitionand the respective statements in ℕ the following

Theorem 1.14 Let a,b,c,d ∈ ℤ.

(1) If a ≤ b, c ≤ d, then a + c ≤ b + d.

(2) If a ≤ b, 0 ≤ c, then ac ≤ bc.

(3) If a ≤ b, c < 0, then bc ≤ ac.

Remarks 1.15 (1) Instead of a < b we also write b > a (“b is bigger than a”) and

in-stead of a ≤ b also b ≥ a (“b is equal or bigger than a”).

(2) We now want to consider the first algebraic property of ℤ We know that ℤ is acommutative ring with unity 1 We call a commutative ring with unity 1 ≠ 0 an

integral domain or just a domain, if there are no nontrivial zero divisors in R, that

is, if a,b ∈ ℝ ⧵ {0} then ab ≠ 0 We have the following.

Theorem 1.16 ℤ is an integral domain.

Proof Let a,b ∈ ℤ, a ≠ 0 ≠ b Then ab ≠ 0.

1.4 The rational numbers ℚ

We also may construct the rational numbers from the integers with the help of anequivalence relation We are guided here by the known representation as fractions

This relation is an equivalence relation Certainly, (a,b) ∼ (a,b) and (c,d) ∼ (a,b)

if (a,b) ∼ (c,d), that is, the relation is reflexive and symmetric It is also transitive, that

is, if (a,b) ∼ (c,d) and (c,d) ∼ (e,f ) then (a,b) ∼ (e,f ) because from ad = bc and cf = ed

we get af = eb Using that ℤ is an integral domain.

Let ℚ ∶=A/∼be the set of equivalence classes for this relation We writea

This last equation holds in ℤ because ab′=a′b and cd′=c′d Therefore the addition

and the multiplication are well defined With this ℚ becomes a field

Remark 1.17 We recall that a field is a set K ≠ ∅ equipped with two binary operations

+ ∶K × K → K and ⋅ ∶ K × K → K satisfying the following three sets of properties:

– K is a commutative group under addition.

– K∗=K ⧵ {0} is a commutative group under multiplication.

– The multiplication is distributive with respect to the addition, that is,

a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) for all a,b,c ∈ K.

The multiplicative inverse fora b , a ≠ 0 ≠ b, is b a

1.5 The absolute value in ℕ, ℤ and ℚ | 15

φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b)

for a,b ∈ ℤ Hence, φ is an embedding, and we may identify ℤ with φ(ℤ), and consider

from now on ℤ as a subset of ℚ We also may transfer the ordering in ℤ to an ordering

This extends the ordering for ℤ to an ordering for ℚ Theorem 1.14 can be directly

transferred from ℤ to ℚ, and we get the following (we now write x instead of a b)

Theorem 1.18 Let x,y,u,v ∈ ℚ.

(1) If x ≤ y, u ≤ v, then x + u ≤ y + x.

(2) If x ≤ y, 0 ≤ u, then xu ≤ yu.

(3) If x ≤ y, u < 0, then yu ≤ xu.

Remarks 1.19 (1) Again, we also write x > y and x ≥ y if y < x and y ≤ x, respectively.

(2) Certainly ℚ is the disjoint union ℚ = ℚ−∪ {0}∪ℚ+, where ℚ−= {x ∈ ℚ ∣ x < 0} and

ℚ+= {x ∈ ℚ ∣ x > 0}.

1.5 The absolute value in ℕ, ℤ and ℚ

We need to consider only ℚ because ℕ ⊂ ℤ ⊂ ℚ

Let x ∈ ℚ The absolute value |x| of x is defined by

|x| ={{

{

x if x ≥ 0,

−x if x < 0.

Lemma 1.20 We always have x ≤ |x|.

Proof If x ≥ 0, then x = |x| If x < 0, then −x > 0, that is, |x| = −x > 0 and therefore

x < 0 < |x|.

Theorem 1.21 Let x,y ∈ ℚ Then the following hold:

(1) |x| ≥ 0 and |x| = 0 ⇔ x = 0.

(2) |x ⋅ y| = |x| ⋅ |y| and |−x| = |x|.

(3) |x + y| ≤ |x| + |y| (triangle inequality).

Proof We have (1) and (2) as an easy exercise We now prove (3) If x + y ≥ 0 then

(2) We have |x| = |x − y + y| ≤ |x − y| + |y|, that is, |x| − |y| ≤ |x − y| Analogously

|y| − |x| ≤ |x − y| Together we get ||x| + |y|| ≤ |x − y|.

Exercises

1 Prove from Theorem 1.2 part (1), (2), (4) and (5), which are:

(1) The addition for ℕ is associative, that is,

a + (b + c) = (a + b) + c,

and commutative, that is,

a + b = b + a.

This means, ℕ is a semigroup with respect to the addition

(2) The multiplication for ℕ is associative, that is,

Exercises | 17

2 Prove n2>2n + 1 for all n ∈ ℕ and n ≥ 3, with the first induction principle.

3 Use induction to prove for any natural number n

a n+1+a n if a n+1⋅a nis even,

a n+1−a n if a n+1⋅a nis odd

Search for formulas to describe a n directly from n, without calculating a m for

m < n and prove these formulas using the first induction principle.

(Hint: By testing one suppose that a 3n−2=4n − 3, a 3n−1=2 and a 3n=4n − 1.)

5 Prove that the addition and the multiplication of the pairs in ℕ × ℕ induce a welldefined addition and multiplication on ℤ, that is, if

(a,b) ∼ (a′,b′) and (c,d) ∼ (c′,d′),then

(a,b) + (c,d) ∼ (a′,b′) + (c′,d′)and

(a,b) ⋅ (c,d) ∼ (a′,b′) ⋅ (c′,d′)

6 Prove that the multiplication on ℚ is well defined, that is, if

(a,b) ∼ (a′,b′) and (c,d) ∼ (c′,d′),then

2 Division and factorization in the integers

2.1 The Fundamental Theorem of Arithmetic

The most important result concerning the integers is the Fundamental Theorem of

Arithmetic This states that any integer can be expressed uniquely as a product of

prime numbers where uniqueness is up to ordering and algebraic sign To give thisimportant result we must define divisibility

In what follows, a,b,c,… are always integers.

Definition 2.1 The integer a is called a divisor or factor of b, written as a ∣ b if there

exists a k ∈ ℤ with b = k ⋅ a We call b a multiple of a.

If a is not a divisor of b, then we write a ∤ b.

The following properties hold:

(1) a ∣ b and b ∣ c ⇒ a ∣ c (transitivity).

(2) c ∣ a and c ∣ b ⇒ c ∣ (k1a + k2b) for all k1,k2∈ ℤ

(3) ±a ∣ a and ±1 ∣ a for all a ∈ ℤ.

as in (2) is called a linear combination of a and b.

More generally, if a1,a2, … ,a n∈ ℤ, then an arithmetic expression

k1a1+k2a2+ ⋯ +k n a n

is called a linear combination of a1,a2, … ,a n

Note that each a with |a| ≥ 2 has at least two positive divisors, that is, 1 and |a|.

Definition 2.2 The integer p ≥ 2 is called a prime number or a prime, if p has exactly

two positive divisors, that is, 1 and p If p is a prime number and p ∣ a, then p is called

a prime divisor or prime factor of a.

Theorem 2.3 Each a ∈ ℕ with a ≥ 2 has a prime factor.

Proof We use the second induction principle, that is, course of values induction The

lowest level is a = 2 The number 2 is a prime so the statement is true at the lowest level Suppose that every integer 2 ≤ k < n has a prime factor, we must show that n

then also has a prime factor

If n is prime then we are done Suppose that n is not a prime number Then n =

m1m2with 1 < m1<n, 1 < m2<n By the inductive hypothesis both m1and m2have

prime factors Therefore n also has a prime factor completing the proof.

Now that we have a prime divisor the next result shows that for any integer theremust be a smallest prime divisor

Theorem 2.4 Each a with |a| ≥ 2 has a smallest prime divisor.

Proof Let T = {x ∣ x ≥ 2 and x ∣ a} Since |a| ∣ a and |a| ≥ 2 we get T ≠ ∅ T contains

a smallest element p ≥ 2 (see Theorem 1.7); p must be a prime number Otherwise if

p = cd with |c| ≥ 2, |d| ≥ 2, then |c| ∣ a, which contradicts the minimality of p because

2 ≤ |c| < p Hence, p is the smallest prime divisor of a.

Definition 2.5 The integer a is called a composite number if there are b and c with

a = bc and 2 ≤ |a|, |b| < |a|.

Theorem 2.6 Let n be a composite number Then n has a prime divisor p with p ≤ √|n|.

Proof Let n = ab with 2 ≤ |a|, |b| < |n| Then |a| ≤ √|n| or |b| ≤ √|n|, because otherwise

|n| = |a||b| > √|n|√|n| = |n|.

Let |a| ≤ √|n| and p be the smallest prime divisor of |a| Then p ∣ n and p ≤ |a| ≤ √|n|.

Remarks 2.7 (1) There are historical reasons to assume that prime numbers are

pos-itive The negative numbers were not settled or known in ancient cultures

Nowa-days we often call p and −p prime elements of ℤ if p is a prime number.

(2) If n is a natural number then one may ask in an obvious manner if n is a prime

number or not A test to determine if a given natural number is prime or not is

called a primality test In this chapter we will not consider primality tests, but at

least we want to mention the deterministic but inefficient Sieve of Eratosthenes.This is a straightforward sieving method to obtain all the primes less than or equal

to a fixed bound x It is ascribed by Eratosthenes (276–194 BC), who was the chief

librarian of the great ancient library in Alexandria

The Sieve of Eratosthenes is a direct method to determine primes It works as lows Given x > 0, list all the positive integers less than or equal to x Starting

fol-with 2, which is a prime number, cross out all multiples of 2 on the list The next

2.1 The Fundamental Theorem of Arithmetic | 21

number on the list not crossed out, which is 3, is a prime number Next cross outall the multiples of 3 not already eliminated The next number left uneliminated,which is 5, is a prime number Continue in this manner As explained in Theo-rem 2.6, for the primality test, the elimination must only be done for numbers less

than or equal √x If, for instance, we take x = 100 then we get as described that

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97are exactly the prime numbers between 2 and 100

We now present and prove the Fundamental Theorem of Arithmetic

Theorem 2.8 (Fundamental Theorem of Arithmetic) Each integer a ≠ 0 can be

writ-ten in a unique way as a product of prime numbers:

a = ϵ∏n

i=1 p i, ϵ = ±1, n ≥ 0, where p1,p2, … ,p n are prime numbers.

The uniqueness is up to the ordering of the factors and the algebraic sign.

Proof If a > 0 then ϵ = +1, and if a < 0 then ϵ = −1 We therefore may assume that

a > 0.

If a = 1, then n = 0, the empty product Hence, let now a ≥ 2.

Existence: We assume that there exists an a ≥ 2 which does not have such a

de-composition Then there exists a smallest such number, which we call b The number

b has a prime divisor p and hence, b = pc with 1 ≤ c < b We have c ≠ 1 because

oth-erwise b = p a prime number, and we have a decomposition of b Therefore 2 ≤ c < b, and c has a decomposition c = p1p2⋯p k Then b = p⋅p1p2⋯p k, which contradicts our

assumption Therefore each a ≥ 2 has such a decomposition.

Uniqueness: We assume that there exists an a ≥ 2 which has such a decomposition

but this is not unique (up to the ordering of the factors)

Then there exists a smallest such number, which we again call b This b has a smallest prime divisor, say q We write b = cd with c ≥ 2, d ≥ 1.

If q ∣ c, then c = fq for some f ∈ ℕ and b = qfd, and q occurs in the decomposition

of b Now, let q ∤ c Let p be a prime divisor of c Then p ∣ c and therefore p ∣ b Because

of the minimality of q and because q ∤ c we necessarily have p > q It follows c > q because p ∣ c We form the number

r = b − qd = (c − q)d ∈ ℕ.

By our assumption r < b Now q ∣ b and q ∣ qd, therefore, q ∣ (c−q)d We have q ∤ (c−q) since otherwise q ∣ (c − q + q) hence q ∣ c Because of the uniqueness for the decompo- sition of r = (c − q)d we must have q ∣ d Hence, also in the case q ∤ c the prime factor

q occurs in each decomposition of b Therefore, altogether, q occurs in each

decom-position of b, that means, if p1,p2, … ,p k are prime numbers with b = p1p2⋯p kthen

q = p i for some i Since the unique decomposition of b

q <b we get the uniqueness for

b = q ⋅ b q which contradicts our assumption The statement now follows from the ond induction principle

sec-Remark 2.9 In general we combine equal prime numbers and write

a = p α1

1 p α2

2 ⋯p α k

k for a ≥ 2, with p1,p2, … ,p k pairwise different prime numbers, k ≥ 1, α1,α2, … ,α k∈ ℕ

Often we sort the p iby size, that is,

p1<p2< ⋯ <p k.The corresponding expression

a = p α1

1 p α2

2 ⋯p α k

k for a ≥ 2

is called the standard prime decomposition of a.

If it is convenient we also write

a = ∏

p p α(p),where the product is over all prime numbers, all α(p) ≥ 0, and almost all α(p) = 0, that

is, all α(p) = 0 up to finitely many α(p) which are possibly not 0.

Corollary 2.10 There exist infinitely many prime numbers.

Proof Assume that there are only finitely many prime numbers

p1,p2, … ,p k, k ≥ 1.

We form

n = p1p2⋯p k+1;

n has a prime divisor p because n ≥ 2 since there exists at least the prime number 2.

Since there are only finitely many prime numbers p1,p2, … ,p k, the prime number

p must occur in the list Hence, without loss of generality, let p = p1 Then n = p1a for

some a ∈ ℕ because p1∣n It follows:

1 = n − p1p2⋯p k=p1(a − p2⋯p k),

which means p1∣1, and this gives a contradiction because p1≥2 Therefore there existinfinitely many prime numbers

2.2 The division algorithm and the greatest common divisor | 23

Corollary 2.11 (Euclid’s Lemma – first version) Let p be a prime number with p ∣ ab,

a,b ∈ ℕ Then p ∣ a or p ∣ b.

Proof If we would have p ∤ a and p ∤ b, then ab would have a prime decomposition

as a product of prime factors of a and b, which does not contain p On the other side

we have p ∣ ab, that is, ab = pc for some c ∈ ℤ If we multiply the decomposition of c with p we get a decomposition of ab = pc which contains p This gives a contradiction

to Theorem 2.8 Therefore, p ∣ a or p ∣ b.

Remark 2.12 If Euclid’s Lemma, which is named after Euclid of Alexandria (ca.

300 BC), is already available, then we get the uniqueness part in Theorem 2.8 for free:

If a ≥ 2 is not a prime number, then a = bc with 1 < b,c < a Let p be a prime visor of a By Euclid’s Lemma then p ∣ b or p ∣ c, and hence, p occurs in each prime decomposition of a which leads via a = p ⋅ a pto the uniqueness

di-What we want is a proof of Euclid’s Lemma without the Fundamental Theorem of

Arithmetic This can be done with the division algorithm that we introduce in the next

section

2.2 The division algorithm and the greatest common divisor

The division algorithm describes the division of an integer by a smaller integer to obtain

a quotient and a remainder

Theorem 2.13 (Division algorithm) Let a,b ∈ ℤ with b ≠ 0 Then a can be presented

in the form

a = qb + r with 0 ≤ r < |b|.

The number q is called the quotient and the number r the remainder This presentation

is unique, that is, q and r are uniquely determined by a and b.

Proof Suppose first that 0 < b The existence of a representation a = qb+r with 0 ≤ r <

b comes from the biggest multiple of b which is less than or equal to a This certainly

exists if a ≥ 0 (see Chapter 1) If a < 0 then −a > 0, and there exists a q ∈ ℤ with (q−1)b <

−a ≤ qb, and hence (−q)b ≤ a < −(q − 1)b, and −qb is the biggest multiple of b which is

less than or equal to a.

For the uniqueness we take two such representations

a = q1b + r1, 0 ≤ r1<b and

a = q2b + r2, 0 ≤ r2<b.

Suppose, without loss of generality, that r1≥r2 Then 0 ≤ r1−r2<b Subtraction leads

to

b(q1−q2) +r1−r2=0,that is,

0 ≤ r1−r2=b(q2−q1) <b.

From this necessarily q1=q2and then r1=r2

Now, let b < 0 Then −b > 0, and there exist unique numbers q1and r1such that

a = q1(−b) + r1 with 0 ≤ r1< −b Therefore a = (−q1)b + r1 and 0 ≤ r1< |b| Now, take

q = −q1and r = r1

Examples 2.14 (1) 87 = 8 ⋅ 10 + 7,

(2) −52 = (−5) ⋅ 12 + 8

Now we want to introduce the greatest common divisor of two integers Later we

will introduce the greatest common divisor of integers a1,a2, … ,a nin general We firstgive a definition which is suitable to present in school and then show the equivalence

of this definition to the more suitable definition for algebra

We introduce some notation If a ∈ ℤ then we write

Definition 2.16 Let a,b ∈ ℤ with a ≠ 0 or b ≠ 0 The greatest common divisor gcd(a,b)

of a and b is

gcd(a,b) ∶= max(T a∩T b),that is, the biggest natural number in T a∩T b

For a = b = 0 the greatest common divisor is not defined.

2.2 The division algorithm and the greatest common divisor | 25

Examples 2.17 (1) T12∩T−15= {1,3}, and hence, gcd(12,−15) = 3

(2) T0∩T12=T12, and hence, gcd(0,12) = 12

Theorem 2.18 Let a,b ∈ ℤ with a ≠ 0 or b ≠ 0 Then, d ∈ ℕ is the gcd(a,b) if and only

if the two conditions are satisfied:

(i) d ∣ a and d ∣ b.

(ii) If δ ∣ a and δ ∣ b for δ ∈ ℕ, then δ ∣ d.

Proof If the two conditions (i) and (ii) hold, then certainly d = gcd(a,b).

Now, let d = gcd(a,b) Then condition (i) is clear by definition.

Assume that condition (ii) does not hold, that is, δ ∣ a and δ ∣ b but δ ∤ d Then there exists a prime number p with p α∣a and p α∣b for some α ∈ ℕ but p α∤d Now,

from d ∣ a and d ∣ b we get p α d ∣ a and p α d ∣ b which contradicts the maximality of d.

Hence δ ∣ d, and condition (ii) holds.

From the definition we obtain automatically the following properties:

gcd(a,1) = 1, gcd(a,b) = gcd(a,−b) = gcd(−a,b) = gcd(−a,−b) = gcd(b,a) for integers a,b, not both 0 Hence, to calculate the gcd(a,b) we may assume that a and b are both non-negative.

Definition 2.19 Let a,b ∈ ℤ, a ≠ 0 or b ≠ 0 The integers a and b are called relatively

prime if gcd(a,b) = 1.

We show first how to calculate the gcd(a,b) in terms of the prime decompositions.

Theorem 2.20 Let a,b ∈ ℕ0= ℕ ∪ {0}, a ≠ 0 or b ≠ 0 If a = 0 then gcd(a,b) = b and

if b = 0 then gcd(a,b) = a Now suppose that a,b > 0 and we consider the prime positions of a and b:

decom-a = p α1

1 p α2

2 ⋯p α n

n,and

n with δ i=min(α i,β i) for i = 1,2,…,n.

Proof Given the prime decompositions of a and b it follows that each common divisor

of a and b is of the form

p γ1

1p γ2

2 ⋯p γ n

n with 0 ≤ γ i≤min(α i,β i) for i = 1,2,…,n.

The result then follows

Example 2.21 Let us find the gcd(12,30) Here

12 = 22⋅31⋅50 and 30 = 21⋅31⋅51.

It follows that gcd(12,30) = 21⋅31⋅50=6

Theorem 2.20 provides an easy way to calculate the gcd(a,b) if we know the prime factorization of a and b In general, however, it is difficult to find those prime factoriza- tions A more efficient method to find the greatest common divisor uses the Euclidean

algorithm that we give in the next section.

2.3 The Euclidean algorithm

The Euclidean algorithm is the process given in Theorem 2.24 It will give a general

method to calculate the gcd(a,b).

First we need the following result Let a,b ∈ ℕ Then by Theorem 2.13 there exists

q and r ∈ ℤ with a = qb + r and 0 ≤ r < b.

Lemma 2.22.

gcd(a,b) = gcd(b,r).

Proof Since r = a − qb each common divisor of a and b is also a divisor of r On the

other side each common divisor of b and r is also a divisor of a Hence,

T a∩T b=T b∩T r,and

gcd(a,b) = gcd(b,r).

Corollary 2.23 From Lemma 2.22 we see that gcd(a,b) = b if b ∣ a.

Theorem 2.24 (Euclidean algorithm) Let a,b ∈ ℕ, a > b and b ∤ a Using the division

algorithm we form the following scheme by repeating divisions:

2.3 The Euclidean algorithm | 27

Proof Since the remainders are non-negative and become smaller in each step it

fol-lows that we must get a remainder 0 after finitely many steps

From Lemma 2.22 we get

Remark 2.26 For the scheme we certainly also may start to divide the smaller number

by the bigger one In this case the first step just gives a permutation of a and b Further it follows from the procedure that the last nonzero remainder r ncan be

expressed as a linear combination of a and b by successively eliminating the r i’s in

the intermediate equations To express r n as a linear combination of a and b notice

first that

r n=r n−2−q n r n−1.Substituting this in the immediately preceding division, we get

r n=r n−2−q n(r n−3−q n−1 r n−2) = (1 + q n q n−1)r n−2−q n r n−3

Doing this successively, we ultimately express r n as a linear combination of a and b.

This leads directly to the following

Theorem 2.27 (Lemma of E Bézout (1730–1783)) Let a,b,d be nonzero integers The

equation

ax + by = d

is solvable with x,y ∈ ℤ if and only if gcd(a,b) ∣ d.

Proof Let ax + by = d for x,y ∈ ℤ If y = 0, then x ≠ 0 because d ≠ 0, a ∣ d and

gcd(a,b) ∣ d because gcd(a,b) ∣ a Analogously we argue with x = 0 Now, let x ≠ 0 ≠ y Then certainly gcd(a,b) ∣ d.

Now, assume that gcd(a,b) ∣ d Then d = gcd(a,b) ⋅ t for some t ∈ ℤ, t ≠ 0, because

d ≠ 0.

Then there exist from the Euclidean algorithm x1,y1∈ ℤwith ax1+by1=gcd(a,b),

and we get

ax1t + by1t = gcd(a,b) ⋅ t = d.

Bézout’s Lemma leads to the following alternative characterization of the gcd(a,b).

Corollary 2.28 The greatest common divisor of two integers a,b is the smallest positive

linear combination of a and b.

Corollary 2.29 Two integers a,b are relatively prime if and only if 1 can be written as

a linear combination of a and b.

Example 2.30 We saw that gcd(525,231) = 21 We show how to express 21 as a linear

Remark 2.31 Let a,b ∈ ℤ, a ≠ 0 or b ≠ 0 It is possible to calculate algorithmically

the gcd(a,b) and the corresponding linear combination simultaneously This is called often the extended Euclidean algorithm Consider the following steps:

(1) If a = 0 and b ≠ 0 then output (0,|b b|, |b|), if a ≠ 0 and b = 0 then output (|a a|,0,|a|)

2.3 The Euclidean algorithm | 29

This is an algorithm which calculates a triple (c,d,e) with

e = gcd(a,b) and e = c ⋅ a + d ⋅ b.

This we can see as follows:

Let a ≠ 0 ≠ b (step (1) with a = 0 or b = 0 is clear) Certainly the procedure stops after finitely many steps because r < e1 The procedure is correct If we consider only

the third components e i, then the usual Euclidean algorithm proceeds Hence, it is

e = gcd(a,b) In each step we have

Corol-Theorem 2.32 (Euclid’s Lemma – second version) Let a,b,c ∈ ℤ, a ≠ 0, b ≠ 0, c ≠ 0.

Let a ∣ bc and gcd(a,b) = 1 Then a ∣ c.

Proof Since gcd(a,b) = 1 there exist x,y ∈ ℤ with ax + by = 1 Then acx + bcy = c Now,

a ∣ a and a ∣ bc, and hence, a ∣ c.

Almost the same proof provides the following corollary

Corollary 2.33 Let a,b,c ∈ ℤ, a ≠ 0, b ≠ 0, c ≠ 0 Let a ∣ c, b ∣ c and gcd(a,b) = 1 Then

ab ∣ c.

Remark 2.34 We now consider the equation ax + by = d, d = gcd(a,b), a,b ∈ ℤ,

a ≠ 0 ≠ b, and ask for all the possible solutions x,y ∈ ℤ.

Without any loss of generality we may assume that d = 1, because if d > 1 then we may divide by d and get the equation a d x + b d y = 1.