algebra and number theory a selection of highlights pdf

Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman Algebra and Number Theory De Gruyter Textbook www.EngineeringBooksPDF.com Also of Interest 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, Gerhard Rosenberger, 2015 ISBN 978-3-11-037276-2, e-ISBN (PDF) 978-3-11-037277-9, e-ISBN (EPUB) 978-3-11-038616-5 Abstract Algebra An Introduction with Applications Derek J S Robinson, 2015 ISBN 978-3-11-034086-0, e-ISBN (PDF) 978-3-11-034087-7, e-ISBN (EPUB) 978-3-11-038560-1 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, 2011 ISBN 978-3-11-025008-4, e-ISBN (PDF) 978-3-11-025009-1 www.EngineeringBooksPDF.com Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman Algebra and Number Theory | A Selection of Highlights www.EngineeringBooksPDF.com Mathematics Subject Classification 2010 0001, 00A06, 1101, 1201 Authors Prof Dr Benjamin Fine Fairfield University Department of Mathematics 1073 North Benson Road Fairfield, CT 06430 USA Prof Dr Gerhard Rosenberger University of Hamburg Department of Mathematics Bundesstr 55 20146 Hamburg Germany Prof Dr Anthony Gaglione United States Naval Academy Department of Mathematics 212 Blake Road Annapolis, MD 21401 USA Prof Dr Dennis Spellman Temple University Department of Mathematics 1801 N Broad Street Philadelphia, PA 19122 USA Dr Anja Moldenhauer University of Hamburg Department of Mathematics Bundesstr 55 20146 Hamburg Germany 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 www.EngineeringBooksPDF.com Preface To many students, as well as to many teachers, mathematics seems like a mundane discipline, filled with rules and algorithms and devoid of beauty and art However to someone who truly digs deeply into mathematics this is quite far from the truth The world 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 gems and 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, eiπ + = 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 if one 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 amazing discipline In this book, which we call Algebra and Number Theory, we introduce and examine many of these exciting results We planned this book to be used in courses for teachers and for the general mathematically interested so it is somewhat between a textbook and just a collection of results We examine these mathematical gems 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 Masters Level Book for prospective teachers With the increasing demand for education in the STEM subjects, there is the realization that to get better teaching in mathematics, the prospective teachers must both be more knowledgeable in mathematics and excited about the subject The courses in teacher preparation not touch many of these results that make the discipline so exciting This book is intended to address this issue The first volume is on Algebra and Number Theory We touch on numbers and number systems, polynomials and polynomial equations, geometry and geometric constructions These parts are somewhat independent so a professor can pick and choose the areas to concentrate on Much more material is included than can be covered in a single course We prove all relevant results that are not too technical or complicated to scare the students We find that mathematics is also tied to its history so we include many historical comments We try to introduce all that is necessary however we presuppose certain subjects from school and undergraduate mathematics These include basic knowledge in algebra, geometry and calculus as well as some knowledge of matrices and linear equations Beyond these the book is self-contained This first volume of two is called Algebra and Number Theory There are fourteen chapters and we think we have introduced a very wide collection of results of the type that we have alluded to above In Chapters 1–5 we look at highlights on the integers We examine unique factorization and modular arithmetic and related ideas We show how these become critical components of modern cryptography especially public key crypDOI 10.1515/9783110516142-201 www.EngineeringBooksPDF.com VI | Preface tographic methods such as RSA Three of the authors (Fine, Moldenhauer and Rosenberger) work partly as cryptographers so cryptography is mentioned and explained in several places In Chapters and we look at exceptional classes of integers such as the Fibonacci numbers as well as the Fermat numbers, Mersenne numbers, perfect numbers and Pythagorean triples We explain the golden section as well as expressing integers as sums of squares In Chapters 6–8 we look at results involving polynomials and polynomial equations We explain field extensions at an understandable level and then prove the insolvability of the quintic and beyond The insolvability of the quintic 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 leading eventually 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 and uses the material we developed on field extensions to prove the impossibility of certain constructions Finally in Chapter 14 we look at Euclidean Vector Spaces We give several geometric applications and look for instance at a secret sharing protocol using the closest vector theorem We would like to thank the people who were involved in the preparation of the manuscript Their dedicated participation in translating and proofreading are gratefully acknowledged In particular, we have to mention Anja Rosenberger, Annika Schürenberg and the many students who have taken the respective courses in Dortmund, Fairfield and Hamburg Those mathematical, stylistic, and orthographic errors that undoubtedly remain shall be charged to the authors Last but not least, we thank de Gruyter for publishing our book Benjamin Fine Anthony Gaglione Anja Moldenhauer Gerhard Rosenberger Dennis Spellman www.EngineeringBooksPDF.com Contents Preface | V 1.1 1.2 1.3 1.4 1.5 The natural, integral and rational numbers | Number theory and axiomatic systems | The natural numbers and induction | The integers ℤ | 10 The rational numbers ℚ | 13 The absolute value in ℕ, ℤ and ℚ | 15 2.1 2.2 2.3 2.4 2.5 Division and factorization in the integers | 19 The Fundamental Theorem of Arithmetic | 19 The division algorithm and the greatest common divisor | 23 The Euclidean algorithm | 26 Least common multiples | 30 General gcd’s and lcm’s | 33 3.1 3.2 3.3 3.4 3.5 Modular arithmetic | 39 The ring of integers modulo n | 39 Units and the Euler φ-function | 43 RSA cryptosystem | 46 The Chinese Remainder Theorem | 47 Quadratic residues | 54 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 4.3 Exceptional numbers | 61 The Fibonacci numbers | 61 The golden rectangle | 67 Squares in semicircles | 68 Side length of a regular 10-gon | 69 Construction of the golden section α with compass and straightedge from a given a ∈ ℝ, a > | 70 Perfect numbers and Mersenne numbers | 71 Fermat numbers | 78 5.1 5.2 5.3 Pythagorean triples and sums of squares | 83 The Pythagorean Theorem | 83 Classification of the Pythagorean triples | 85 Sum of squares | 89 www.EngineeringBooksPDF.com VIII | Contents 6.1 6.2 6.3 6.3.1 6.3.2 6.4 6.5 6.5.1 6.5.2 6.5.3 6.6 6.6.1 6.6.2 6.6.3 Polynomials and unique factorization | 95 Polynomials over a ring | 95 Divisibility in rings | 98 The ring of polynomials over a field K | 100 The division algorithm for polynomials | 101 Zeros of polynomials | 103 Horner-Scheme | 108 The Euclidean algorithm and greatest common divisor of polynomials over fields | 112 The Euclidean algorithm for K[x] | 114 Unique factorization of polynomials in K[x] | 115 General unique factorization domains | 116 Polynomial interpolation and the Shamir secret sharing scheme | 117 Secret sharing | 117 Polynomial interpolation over a field K | 117 The Shamir secret sharing scheme | 121 7.1 7.2 7.3 7.3.1 7.4 Field extensions and splitting fields | 125 Fields, subfield and characteristic | 125 Field extensions | 126 Finite and algebraic field extensions | 131 Finite fields | 134 Splitting fields | 135 8.1 8.2 8.2.1 8.2.2 8.3 Permutations and symmetric polynomials | 141 Permutations | 141 Cycle decomposition of a permutation | 144 Conjugate elements in Sn | 147 Marshall Hall’s Theorem | 148 Symmetric polynomials | 151 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.7.1 9.7.2 9.7.3 Real numbers | 157 The real number system | 157 Decimal representation of real numbers | 168 Periodic decimal numbers and the rational number | 172 The uncountability of ℝ | 173 Continued fraction representation of real numbers | 175 Theorem of Dirichlet and Cauchy’s Inequality | 176 p-adic numbers | 178 Normed fields and Cauchy completions | 179 The p-adic fields | 180 The p-adic norm | 183 www.EngineeringBooksPDF.com Contents | IX 9.7.4 9.7.5 10 10.1 10.2 10.2.1 10.2.2 10.2.3 10.2.4 10.3 10.3.1 10.3.2 10.4 10.5 The construction of ℚp | 184 Ostrowski’s theorem | 185 The complex numbers, the Fundamental Theorem of Algebra and polynomial equations | 189 The field ℂ of complex numbers | 189 The complex plane | 193 Geometric interpretation of complex operations | 196 Polar form and Euler’s identity | 197 Other constructions of ℂ | 201 The Gaussian integers | 201 The Fundamental Theorem of Algebra | 202 First proof of the Fundamental Theorem of Algebra | 204 Second proof of the Fundamental Theorem of Algebra | 207 Solving polynomial equations in terms of radicals | 209 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 e and π throughout mathematics | 263 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 13.1 13.2 Compass and straightedge constructions and the classical problems | 289 Historical remarks | 289 Geometric constructions | 289 www.EngineeringBooksPDF.com 318 | 14 Euclidean vector spaces after J P Gram (1850–1916) and E Schmidt (1876–1959) to find this orthogonal basis starting with any basis Finally we use this orthogonal basis to solve the closest vector problem This is given a subspace W of an Euclidean vector space V and a v⃗ ∈ V to find the vector in W closest to v.⃗ In the subsequent two sections we will present nice applications of the closest vector theorem In the following let V be an Euclidean vector space with a scalar product ⟨ , ⟩ Definition 14.23 A set of vectors v1⃗ , v2⃗ , … , vn⃗ ∈ V is called an orthogonal set, if ⟨vi⃗ , vj⃗ ⟩ = for i ≠ j, that is, if any two different vectors of the v1⃗ , v2⃗ , … , vn⃗ are orthogonal This set of vectors is called an orthonormal set if it forms an orthogonal set and if ‖vi⃗ ‖ = for all i = 1, 2, … , n If {v1⃗ , v2⃗ , … , vn⃗ } is an orthonormal set in V and, in addition, a basis of V , then {v1⃗ , v2⃗ , … , vn⃗ } is called an orthonormal basis of V If V is infinite dimensional, then a subset B ⊂ V is called an orthonormal basis of V if B is a basis of V and if each finite subset of B is an orthonormal set in V Lemma 14.24 An orthogonal set {v1⃗ , v2⃗ , … , vn⃗ }, vi⃗ ≠ 0⃗ for i = 1, 2, … , n, is linearly independent Proof Let c1 v1⃗ + c2 c2⃗ + ⋯ + cn vn⃗ = 0,⃗ all ci ∈ ℝ We must show that c1 = c2 = ⋯ = cn = We have ⟨vi⃗ , c1 v1⃗ + c2 v2⃗ + ⋯ + cn vn⃗ ⟩ = ⟨vi⃗ , 0⟩⃗ = hence, c1 ⟨vi⃗ , v1⃗ ⟩ + c2 ⟨vi⃗ , v2⃗ ⟩ + ⋯ + cn ⟨vi⃗ , vn⃗ ⟩ = for all i = 1, 2, … , n Since ⟨vi⃗ , vj⃗ ⟩ = if i ≠ j, we get ci ⟨vi⃗ , vi⃗ ⟩ = for i = 1, 2, … , n Now, vi⃗ ≠ 0,⃗ that is, ⟨vi⃗ , vi⃗ ⟩ ≠ 0, and therefore ci = for i = 1, 2, … , n Hence, {v1⃗ , v2⃗ , … , vn⃗ } is linearly independent Theorem 14.25 Let {e1⃗ , e2⃗ , … , en⃗ } be an orthonormal basis of V , and if v⃗ ∈ V then n v⃗ = ∑⟨v,⃗ ei⃗ ⟩ei⃗ i=1 Proof Since especially {e1⃗ , e2⃗ , … , en⃗ } is a basis of V , there are c1 , c2 , … , cn ∈ ℝ with v⃗ = c1 e1⃗ + c2 e2⃗ + ⋯ + cn en⃗ It follows ⟨v,⃗ ei⃗ ⟩ = c1 ⟨e1⃗ , ei⃗ ⟩ + c2 ⟨e2⃗ , ei⃗ ⟩ + ⋯ + cn ⟨en⃗ , ei⃗ ⟩ for i = 1, 2, … , n Since ⟨ei⃗ , ej⃗ ⟩ = if i ≠ j and ‖ei⃗ ‖ = √⟨ei⃗ , ei⃗ ⟩ = we get that ⟨v,⃗ ei⃗ ⟩ = ci for i = 1, 2, … , n as desired 14.3 Orthonormalization and closest vector | 319 If v⃗ ∈ V and {e1⃗ , e2⃗ , … , en⃗ } is an orthonormal basis then ci = ⟨v,⃗ ei⃗ ⟩ are called the Fourier coefficients of v⃗ relative to {e1⃗ , e2⃗ , … , en⃗ } We now give a procedure to find an orthogonal basis starting with any basis This is called the Gram–Schmidt Orthogonalization Procedure Theorem 14.26 (The Gram–Schmidt Orthogonalization Procedure) Let V be an Euclidean vector space with scalar product ⟨ , ⟩ Let {x1⃗ , x2⃗ , … , xn⃗ , …} be a linearly independent set in V Then the set {v1⃗ , v2⃗ , … , vn⃗ , …} is an orthogonal set, in which the vi⃗ are defined by v1⃗ = x1⃗ k ⃗ , vi⃗ ⟩ ⟨xk+1 vi⃗ ⟨ v i⃗ , vi⃗ ⟩ i=1 ⃗ = xk+1 ⃗ −∑ and vk+1 for k ≥ The orthogonal set {v1⃗ , v2⃗ , … , vn⃗ , …} is called the Gram–Schmidt-Orthogonalization (GSO) of {x1⃗ , x2⃗ , … , xn⃗ , …} Proof Define v1⃗ ∶= x1⃗ Then v2⃗ = x2⃗ − ⟨x2⃗ , v1⃗ ⟩ v⃗ ⟨v1⃗ , v1⃗ ⟩ It follows ⟨v2⃗ , v1⃗ ⟩ = ⟨x2⃗ , v1⃗ ⟩ − ⟨x2⃗ , v1⃗ ⟩ ⟨v1⃗ , v1⃗ ⟩ = ⟨x2⃗ , x1⃗ ⟩ − ⟨x2⃗ , x1⃗ ⟩ = ⟨v1⃗ , v1⃗ ⟩ This is the start for an induction Now, let j > and assume ⟨vi⃗ , vk⃗ ⟩ = for all i < j and for all k < i We have j−1 vj⃗ = xj⃗ − ∑ i=1 ⟨xj⃗ , vi⃗ ⟩ ⟨vi⃗ , vi⃗ ⟩ vi⃗ If k < j, then j−1 ⟨vj⃗ , vk⃗ ⟩ = ⟨xj⃗ , vk⃗ ⟩ − ∑ i=1 ⟨xj⃗ , vi⃗ ⟩ ⟨vi⃗ , vi⃗ ⟩ ⟨vi⃗ , vk⃗ ⟩ By induction we have ⟨vi⃗ , vk⃗ ⟩ = for k < i < j Hence, ⟨vi⃗ , vk⃗ ⟩ we can unequal zero only if k = i = j − But then ⃗ ⟩− ⟨vj⃗ , vk⃗ ⟩ = ⟨xj⃗ , vj−1 ⃗ ⟩ ⟨xj⃗ , vj−1 ⃗ , vj−1 ⃗ ⟩ ⟨vj−1 ⃗ , vj−1 ⃗ ⟩ = ⟨vj−1 Now, let V be finite dimensional If {x1⃗ , x2⃗ , … , xn⃗ } is a basis of V , then the GSO leads to an orthogonal set {v1⃗ , v2⃗ , … , vn⃗ } which is also a basis of V If we divide the vi⃗ by ‖vi⃗ ‖, then we get an orthonormal basis of V Therefore we have the following 320 | 14 Euclidean vector spaces Theorem 14.27 If V is a finite dimensional Euclidean vector space, then V has an orthonormal basis Example 14.28 Let U ⊂ ℝ4 be the subset of ℝ4 with a basis u⃗1 = (1, 0, 0, 2), u⃗2 = (2, 1, 1, 0) and u⃗3 = (0, 2, 3, 0) We construct an orthogonal basis of U Let v1⃗ = u⃗1 = (1, 0, 0, 2) Then v2⃗ = u⃗2 − ⟨u⃗2 , v1⃗ ⟩ v⃗ = (2, 1, 1, 0) − (1, 0, 0, 2, ) = ( , 1, 1, − ) 5 ⟨v1⃗ , v1⃗ ⟩ Now v3⃗ = u⃗3 − ⟨u⃗3 , v1⃗ ⟩ ⟨u⃗3 , u⃗2 ⟩ − v⃗ , ⟨v1⃗ , v1⃗ ⟩ ⟨v2⃗ , v2⃗ ⟩ and we get v3⃗ = (− Then v⃗ v⃗ v1⃗ , , ‖v1⃗ ‖ ‖v2⃗ ‖ ‖v3⃗ ‖ 40 27 53 20 , , , ) 26 26 26 26 form an orthonormalbasis of U If V is an Euclidean vector space and W ⊂ V is subspace then the closest vector problem is to determine given v⃗ ∈ V the closest vector w⃗ ∈ W to v.⃗ In geometric terms w⃗ is the orthogonal projection of v.⃗ This can be solved in terms of an orthonormal basis of W The solution is called the closest vector theorem Theorem 14.29 (Closest vector theorem) Let W be a subspace of an Euclidean vector space V with scalar product ⟨ , ⟩ Let v⃗ be a vector of V If {e1⃗ , e2⃗ , … , en⃗ } is an orthonormal basis of W then the unique vector w⃗ ∈ W closest to v⃗ is given by n w⃗ = ∑⟨v,⃗ ei⃗ ⟩ei⃗ i=1 Proof Each vector from W can be written uniquely as a linear combination c1 e1⃗ + c2 e2⃗ + ⋯ + cn en⃗ with certain c1 , c2 , … , cn ∈ ℝ We have to find c1 , c2 , … , cn such that ⟨v⃗ − (c1 e1⃗ + c2 e2⃗ + ⋯ + cn en⃗ ), v⃗ − (c1 e1⃗ + c2 e2⃗ + ⋯ + cn en⃗ )⟩ becomes minimal 14.4 Polynomial approximation | 321 Since {e1⃗ , e2⃗ , … , en⃗ } is an orthonormal basis of W , we get ⟨v⃗ − (c1 e1⃗ + c2 e2⃗ + ⋯ + cn en⃗ ), v⃗ − (c1 e1⃗ + c2 e2⃗ + ⋯ + cn en⃗ )⟩ n n i=1 n i=1 = ‖v‖⃗ − ∑ ci ⟨v,⃗ ei⃗ ⟩ + ∑ ci2 n = ‖v‖⃗ + ∑(⟨v,⃗ ei⃗ ⟩ − ci )2 − ∑(⟨v,⃗ ei⃗ ⟩)2 i=1 i=1 Since the second term is non-negative, the whole right side is minimal if the second term is zero, or equivalently, if ⟨v,⃗ ei⃗ ⟩ = ci for all i = 1, 2, … , n Remark 14.30 If v⃗ ∈ W , then this is just the representation of v⃗ as a linear combination in the orthogonal basis {e1⃗ , e2⃗ , … , en⃗ } Example 14.31 Let W ⊂ ℝ4 be the subspace generated by u⃗1 = (1, 0, 0, 2), u⃗2 = (2, 1, 1, 0) and u⃗3 = (0, 2, 3, 0) In Example 14.28 we saw that v1⃗ = (1, 0, 0, 2), v2⃗ = ( , 1, 1, − ) and 5 v3⃗ = (− 40 27 53 20 , , , ) 26 26 26 26 is an orthogonal basis of W By normalization to length we get that 1 , , ,− , 0, 0, ), e2⃗ = ( ) √5 √5 √82 √82 √82 √82 40 27 53 20 e3⃗ = (− , , , ) √5 538 √5 538 √5 538 √5 538 e1⃗ = ( and is an orthonormal basis of W The vector w⃗ ∈ W closest to v⃗ = (0, 1, 0, 1) is given by w⃗ = ⟨v,⃗ e1⃗ ⟩ + ⟨v,⃗ e2⃗ ⟩ + ⟨v,⃗ e3⃗ ⟩ = (−445 348, 437 295, 938 315, 532 204) 268 530 14.4 Polynomial approximation We now give a very nice application of the closest vector theorem to a problem in analysis Given an arbitrary continuous function f on [a, b] the Weierstrass approximation theorem says that f (x) can be uniformly approximated by a polynomial An important problem in analysis is to determine, given a continuous function f (x) on [a, b], the best polynomial approximation for it One solution to this problem is called least squares approximation 322 | 14 Euclidean vector spaces If f (x) is an arbitrary continuous function on [a, b] Then the least squares approximation of degree n for f (x) is the polynomial Pn (x) of degree n or less that minimizes b ∫ |f (x) − Pn (x)|2 dx a The least squares approximation can be found easily using the closest vector theorem Let V be the Euclidean vector space of the continuous functions f ∶ [a, b] → ℝ, a, b ∈ ℝ, a < b, with the scalar product b ⟨f , g⟩ = ∫ f (t)g(t)dt a Let n ∈ ℕ and f ∈ V be given Let W = Rn [x] be the subspace of V given by polynomials of degree less than or equal to n This has a basis 1, x, x2 , … , xn For given f (x) ∈ V the least squares approximation of degree n is the element of W closest to f (x) with respect to the scalar product ⟨ , ⟩ in V We can find this polynomial with the following algorithm A basis of ℝn [x] is given by the polynomials 1, x, x2 , … , xn Step 1: Using the GSO we have to find an orthogonal basis {po (x), p1 (x), … , pn (x)} of ℝn [x] Step 2: By normalization of the orthogonal basis we get an orthonormal basis {φ0 (x), φ1 (x), … , φn (x)} of ℝn [x] For this we have especially b ∫ φi (t)φj (t)dt = δij , a where {1, δij = { 0, { for i = j, for i ≠ j the Kronecker symbol Step 3: Find c0 , c1 , … , cn ∈ ℝ with pn (x) = c0 φ0 (x) + c1 φ1 (x) + ⋯ + cn φn (x) and b ci = ∫ f (t)φi (t)dt a for i = 0, 1, … , n Example 14.32 Let n = and f (x) = ex , x ∈ [0, 1] Step 1: For {1, x, x2 } we find an orthogonal basis Let u1⃗ = 1, u2⃗ = x and u3⃗ = x With the Gram–Schmidt orthogonalization procedure we get v1⃗ = and v2⃗ = u2 − ⟨u⃗2 , v1⃗ ⟩ v⃗ ⟨v1⃗ , v1⃗ ⟩ 14.5 Secret sharing scheme using the closest vector theorem We have ⟨u⃗2 , v1⃗ ⟩ = ∫ tdt = Hence, v2⃗ = x − | 323 1 and ⟨v1⃗ , v1⃗ ⟩ = ∫ dt = Using this we get v3⃗ = u⃗3 − Now, ⟨u⃗3 , v1⃗ ⟩ ⟨u⃗3 , v2⃗ ⟩ v2⃗ − v⃗ ⟨v2⃗ , v2⃗ ⟩ ⟨v1⃗ , v1⃗ ⟩ t2 )dt = , 12 1 ⟨u3⃗ , v1⃗ ⟩ = ∫ t dt = , and 1 ⟨v2⃗ , v2⃗ ⟩ = ∫ (t − ) dt = 12 ⟨u⃗3 , v2⃗ ⟩ = ∫ (t − We get therefore v3⃗ = x2 − x + 61 Step 2: We normalize the Gram–Schmidt orthogonal basis With ‖v1⃗ ‖ = we have φ0 (x) = 1; with ‖v2⃗ ‖ = √1 we have φ1 (x) = √12(x − 21 ); with 12 ‖v3⃗ ‖ = we have φ2 (x) = √180(x2 − x + ) √180 Step 3: We calculate the ⟨ex , φi (x)⟩ for i = 0, 1, It is ⟨ex , φ0 (x)⟩ = ∫ et dt = 1,718 … , 1 ⟨e , φ1 (x)⟩ = ∫ et (√12(t − ))dt = 0,4880 … , 1 x ⟨e , φ2 (x)⟩ = ∫ et (√180(t − t + ))dt = 0,0625 … x For the best approximation we get P2 (x) = ⟨ex , φ0 (x)⟩φ0 (x) + ⟨ex , φ1 (x)⟩φ1 (x) + ⟨ex , φ2 (x)⟩φ2 (x), hence, P2 (x) ≈ 1,718 + 0,4880φ1 (x) + 0,0625φ2 (x) Notation The Gram–Schmidt orthogonal set which belongs to {1, x, x2 , …} on [0, 1] is called the set of the Legendre polynomials on [0, 1] 14.5 Secret sharing scheme using the closest vector theorem As a second application we give an alternative secret sharing scheme using the closest vector theorem Recall that a secret sharing scheme among a group of n participants is a very important cryptographic protocol The general idea is the following We have n people who have access to a secret S in such a way that the secret can be recov- 324 | 14 Euclidean vector spaces ered if any t of the access group with t ≤ n get together Given such a secret S, then an (n, t)-secret sharing (threshold) scheme is a cryptographic primitive in which a secret is split into pieces (shares) and distributed among a collection of n participants {p1 , p2 , … , pn } so that any group of t, t ≤ n, or more participants can recover the secret Meanwhile, any group of t − or fewer participants cannot recover the secret By sharing a secret in this way the availability and reliability issues can be solved A geometric alternative scheme to Shamir’s secret sharing scheme (see Section 6.6.3) uses the Closest Vector Theorem, see Theorem 14.29 Now we explain the secret sharing scheme using the Closest Vector Theorem We start with an Euclidean vector space V of dimension m and an access control group of size n We assume that the dimension m of V is much greater than n, that is, m ≫ n Within V there is a hidden subspace W of dimension t < n The secret to be shared is given as an element in this hidden subspace, that is, the secret v ∈ W a vector in W The dealer distributes to each of the n members of the access control group, i = 1, 2, … , n, two vectors, vi and w, where vi ∈ W , and w is a vector in the big space V The common vector w has the property that w ∉ W and v is the vector in W closest to w In general the vector w can be given publicly The set {v1 , v2 , … , } has the property that any subset of size t is independent Hence any subset of size t determines a basis for W Suppose t valid users get together They can determine a basis for W and hence using the Gram–Schmidt orthogonalization procedure (see Theorem 14.26) they determine an orthonormal basis Since w is given, they can determine v by the Closest Vector Theorem and recover the secret Given a subset of size less than t the given vectors generate a subspace of W of dimension less than t and hence in V there are infinitely many extensions to subspaces of dimension t This implies that determining W with less than t elements of a basis has negligible probability This is a general method like the Shamir protocol In [4], Chum, Fine, Moldenhauer, Rosenberger and Zhang, compared several different secret sharing protocols including the classic Shamir secret sharing scheme and the secret sharing scheme using the Closest Vector Theorem explained above Exercises Let v⃗ = (1, 2, 3) and w⃗ = (6, 5, 4) be two vectors in ℝ3 Calculate the distance between v⃗ and w.⃗ Let v1⃗ = (3, 0, 40) and v2⃗ = (1, 1, 1) be two vectors in ℝ3 Calculate an orthonormal basis for the subspace in ℝ3 , which is generated by v1⃗ and v2⃗ Consider the three vectors x⃗ = (0, 1, 1), in ℝ3 y⃗ = (1, 1, 0), z ⃗ = (1, 0, 1), Exercises | 325 (a) Calculate the distance between the two vectors x ⃗ and y.⃗ (b) Determine an orthonormal basis for the subspace U ⊂ ℝ3 generated by x ⃗ and z.⃗ (c) Calculate the closest vector in U to y.⃗ Let V be an Euclidean vector space and v,⃗ w⃗ ∈ V Show ⟨v,⃗ w⟩⃗ = (‖v‖⃗ + ‖w‖⃗ − ‖v⃗ − w‖⃗ ) Given are two straight-lines g1 and g2 , with g1 ∶ x⃗ = (1, 1) + ℝ(1, 2) and g2 ∶ y⃗ = (3, 1) + ℝ(2, 1) Calculate the angle under which the lines intersect Show that the four sides of a (non-degenerate) parallelogram are equal if the both diagonals intersect perpendicular (Hint: Without loss of generality consider the following situation shown in Fig⃗ ure 14.15 It is a⃗ = v⃗ + w,⃗ b⃗ = −v⃗ + w.) Figure 14.15: (Non-degenerate) parallelogram with equal sides (a) Show that the set of vectors in ℝ4 of the form {(x, 2x, y, x + y)} forms a subspace and determine an orthogonal basis for it (b) Find the dimension of the subspace of ℝ4 spanned by v1⃗ = (3, 6, 3, 0), v2⃗ = (4, 2, 1, 1), v3⃗ = (2, 0, 2, −2) and give a basis for it Then give a general form for a vector in this subspace Find the vector in this subspace closest to (1, 1, 0, 0) Find the vector in the subspace U spanned by u⃗1 = (1, 1, −1, 1), u⃗2 = (3, 2, −1, 0) in ℝ4 closest to v⃗ = (0, 7, 4, 7) (a) Determine an orthonormal basis for the subspace V of ℝ3 spanned by u⃗1 = (1, 2, 3), u⃗2 = (2, 3, 1), u⃗3 = (1, 3, 2) (b) Use the results from (c) to find the vector in V closest to (1, 5, 1) Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] G Baumslag, B Fine, M Kreuzer, and G Rosenberger A Course in Mathematical Cryptography De Gruyter, 2015 W Borho, J C Jantzen, H.-P Kraft, J Rohlfs, and D Zagier Lebendige Zahlen Birkhäuser, 1981 C Carstensen, B Fine, and G Rosenberger Abstract Algebra De Gruyter, 2011 C S Chum, B Fine, A I S Moldenhauer, G Rosenberger, and X Zhang On secret sharing protocols Contemporary Mathematics, 677:51–78, 2016 M Dörfer and G Rosenberger Zeta functions of finitely generated nilpotent groups Groups Korea’94, Eds.: Kim/Johnson, Walter de Gruyter, pages 35–46, 1995 R A Dunlop The Golden Ratio and Fibonacci Numbers World Scientific, 1999 B Fine The Algebraic Theory of Bianchi Groups Marcel Dekker, 1989 B Fine and G Rosenberger The Fundamental Theorem of Algebra Springer, 1997 B Fine and G Rosenberger Algebraic Generalizations of Discrete Groups Marcel Dekker, 2001 B Fine and G Rosenberger Number Theory: An Introduction via the Density of Primes Birkhäuser, 2nd edition, 2016 D Gorenstein, R Lyons, and R Solomon The Classification of the Finite Simple Groups Mathematical Surveys and Monographs, Volumes 40.1–40.6, American Mathematical Society, 1994–2005 M D Greenberg Advanced Engineering Mathematics Prentice Hall, Englewood Cliffs, 1988 F Grunewald, D Segal, and G Smith Subgroups of finite index in nilpotent groups Inventiones Mathematicae, 9:185–223, 1988 B Hornfeck Algebra De Gruyter, 3th edition, 1976 H L Keng Introduction to Number Theory Springer-Verlag, 1982 M Kreuzer and G Rosenberger Growth in Hecke groups Contemporary Mathematics, 629:261–281, 2014 A Mann How Groups Grow London Mathematical Society, Lecture Note Series 395, 2011 G Müller Elementare Zahlentheorie Arithmetik als Prozess, Klett-Verlag, pages 255–290, 2004 I Niven, H S Zuckerman, and H L Montgomery The Theory of Numbers Wiley, 5th edition, 1991 P Ribenboim The Book of Primes Number Records Springer, 1989 S Singh Fermat’s Last Theorem HarperPress, 2012 M J Wiener Cryptoanalysis of Short RSA Secret Exponents IEEE Transaction on Information Theory, 36(3):553–558, 1990 DOI 10.1515/9783110516142-015 Index Abelian group Absolute value 15, 193 Abundant number 76 Algebraic 131 Algebraic closure 227 Algebraic extension 131 Algebraic integer 235 Algebraic number 131, 227, 228 Algebraic number field 227 Algebraically closed 203 Alternating groups An 149 Altitude theorem 314 Amicable number 76 Archimedean 182 Argument 198 Associates 99 Axioms Binomial coefficient 257 Binomial formula 284 Canonical scalar product 304 Cardano’s formulas 212 Cathetus theorem 314 Cauchy complete 165 Cauchy completion 157, 166, 179 Cauchy sequence 157, 195 Cauchy–Schwarz-Inequality 305 Cauchy’s Inequality 177 Characteristic 125 Chinese Remainder Theorem 47 Closest vector theorem 324 Commutative ring with unity 12, 95 Complete normed field 179 Complex conjugate 193 Complex integers 201 Complex numbers 189 Complex plane 196 Composite 99 Composite number 20 Congruence class 39 Congruence modulo n 39 Conjugate 90 Conjugate elements in groups 147 Conjugates 228 Constructible numbers 291 Construction of a regular n-gon 298 Continued fraction 175 Continuum hypothesis 175 Convergent sequence 157 Coprime 25 Coset 149 Cosine rule 310 Cosine rule for ℝ3 316 Countable 173 Course of values induction Cycle multiplication 146 Cyclic group 45 Decimal fraction 169 Decimal numbers 168, 169 Deficient number 76 Degree 98 Discriminant 231, 240 Division algorithm 23 Division ring 220 Divisor 19, 99 Donkey Bridge in ℝ2 312 Doubling the cube 296 Elementary symmetric polynomial 152 Equivalent norms 180 Euclidean algorithm 26 Euclidean domain 202 Euclidean quadratic number field 245 Euclidean vector space (or scalar product space) 303 Euler φ-function 43 Euler’s Formula 200 Euler’s identity 199 Extension by radicals 210 Extension field 126 Extreme to mean ratio 67 Factor 19 Fermat numbers 78 Fermat prime number 78 Fermat’s Big Theorem 87 Fermat’s Little Theorem 42 Fermat’s two-square theorem 89 Ferrari’s formula 209 Fibonacci Numbers 61 Field 14, 95 Field extension 126 330 | Index Field of p-adic numbers 184 Finite extension 128, 131 Finitely generated 131 First induction principle Fourier coefficients 283, 319 Fourier Convergence Theorem 283 Fourier series 283 Fundamental Theorem for Polynomials 115 Fundamental Theorem of Arithmetic 19 Least squares approximation 322 Least upper bound property 166 Least well-ordering property (LWO) Left transversale 149 Legendre polynomials 323 Length 305 Linear combination 19 Liouville-Number 273 Lub property 166 g-nary numbers 169 Galois theory 209 Gaussian integers 201 Gauss’s Lemma 106 Geometric construction 289 Geometric construction problems 289 Golden angle 63 Golden ratio 67 Golden rectangle 67 Golden section 67 Gram–Schmidt Orthogonalization Procedure (GSO) 319 Greatest common divisor (gcd) 24, 113 Group Group of units 99 Mathematical induction Mersenne number 71 Mersenne prime 71 Minimal polynomial 132 Modular group 90 Modulus 198 Monic polynomial 98 Monoid Multiple 99 Multiple zero 135 Hamiltonian skew field 220 Homomorphism 149 Horner-Scheme 109 Ideal 33, 99 Imaginary quadratic field 241 Indeterminate 97 Index 149 Induction Infinite descent 87 Inner product space 303 Integers Integral basis 238 Integral domain 13, 33, 40, 95 Interpolating polynomial 117 Irrational number 173 Irreducible polynomial 100 Isomorphism 127, 149 Lagrange interpolation 118 Law of Cosines 311 Leading coefficient 98 Least common multiple (lcm) 30 n-th root 138 Natural numbers Nested intervals property 168 Non-Archimedean 182 Norm in an algebraic number field 232 Norm on a field 179 Normal subgroup 149 Normed field 179 Order isomorphic 166 Ordered field 161 Orthocenter of a triangle in ℝ2 314 Orthogonal 309 Orthogonal complement 309 Orthogonal set 318 Orthonormal basis 318 Orthonormal set 318 p-adic norm 183 p-adic numbers 184 p-adic valuation 183 Peano Axioms Pell’s equation 244 Perfect numbers 72 Permutation group 142 Polar form 198 Polynomial 97 Polynomial Approximation 321 Index | 331 Polynomial interpolation 117 Primality test 20 Prime 19 Prime divisor 19 Prime element 99 Prime factor 19 Prime field 125 Prime number 19 Primitive element 131, 228 Primitive element theorem 137 Primitive integral polynomial 235 Primitive polynomial 106 Primitive root 45 Principal ideal 99 Principal ideal domain (PID) 33, 113 Principle of mathematical induction Projection 311 Proper divisor 99 Pythagorean triple 83 ℚp 184 Quadratic fields 240 Quadratic integers 241 Quadratic nonresidue modulo n 54 Quadratic residue 89 Quadratic residue modulo n 54 Quaternions 220 Quotient 23 Second principle of induction Secret sharing scheme 117, 323 Secret sharing threshold scheme 117 Semigroup Shamir’s secret sharing scheme 121 Share distribution 121 Sieve of Eratosthenes 20 Simple extension 131 Simple extension by radicals 210 Simple group 149 Simple zero 135 Skew field 220 Sociable number 77 Solvable by radicals 210 Splitting field 125, 135 Squaring the circle 296 Stabilizer 142 Standard prime decomposition 22 Stirling’s approximation 264 Strong triangle inequality 182 Subfield 125 Subring 33 Sum of squares 89 Supremum 166 Symmetric function 151 Symmetric group 141, 142 Symmetric polynomial 151, 152 Ramified 246, 247 Rational Root Theorem 105 Real number system 157 Real numbers 166 Real quadratic field 241 Reducible polynomial 100 Relatively prime 25 Remainder 23 Repeating decimal 172 Residue 39 Residue class 39 Residue class ring 40 Ring 11, 95 Ring of integers modulo n 40 Ring of polynomials 98 Roots of unity 206 RSA Cryptosystem 46 Terminating decimal 172 Theorem Theorem of Abel 216 Theorem of Cayley 143 Theorem of Dirichlet 176 Theorem of Frobenius 224 Theorem of Hall 149 Theorem of Kronecker 130 Theorem of Ostrowski 185 Theorem of Pythagoras 310 Theorem of Thales in ℝ2 312 Trace 233 Transcendental extension 131 Transcendental number 227, 228 Transposition 146 Transversal 149 Triangular number Trisection of an angle 297 Sn 142 Scalar product 303 Ultra-metric 182 Uncountable 173 332 | Index Unique factorization domain (UFD) 116, 202 Unit 43, 99 Unit group 99 Wallis Product Formula 266 Wilson’s theorem 41 Vandermonde determinant 118, 232 Vandermonde interpolation 120 Vandermonde matrix 118 ℤn 40 Zero of a polynomial 104 Zeta function 150 ... complex numbers leading eventually 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... We also have here the formula a1 a2 ⋯ an = gcd (a1 , a2 , … , an ) ⋅ lcm (a1 , a2 , … , an ), and we may calculate lcm (a1 , a2 , … , an ) by lcm (a1 , a2 , … , an ) = a1 a2 ⋯ an gcd (a1 , a2 ,... properties hold: a ∣ b and b ∣ c ⇒ a ∣ c (transitivity) c ∣ a and c ∣ b ⇒ c ∣ (k1 a + k2 b) for all k1 , k2 ∈ ℤ ? ?a ∣ a and ±1 ∣ a for all a ∈ ℤ a ∣ for all a ∈ ℤ ∣ a ⇔ a = a ∣ b and b ∣ a ⇒ a = ±b These