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Field Theory Pete L. Clark
Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9. -
Extending Euclidean Constructions with Dynamic Geometry Software
Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015) Extending Euclidean constructions with dynamic geometry software Alasdair McAndrew [email protected] College of Engineering and Science Victoria University PO Box 18821, Melbourne 8001 Australia Abstract In order to solve cubic equations by Euclidean means, the standard ruler and compass construction tools are insufficient, as was demonstrated by Pierre Wantzel in the 19th century. However, the ancient Greek mathematicians also used another construction method, the neusis, which was a straightedge with two marked points. We show in this article how a neusis construction can be implemented using dynamic geometry software, and give some examples of its use. 1 Introduction Standard Euclidean geometry, as codified by Euclid, permits of two constructions: drawing a straight line between two given points, and constructing a circle with center at one given point, and passing through another. It can be shown that the set of points constructible by these methods form the quadratic closure of the rationals: that is, the set of all points obtainable by any finite sequence of arithmetic operations and the taking of square roots. With the rise of Galois theory, and of field theory generally in the 19th century, it is now known that irreducible cubic equations cannot be solved by these Euclidean methods: so that the \doubling of the cube", and the \trisection of the angle" problems would need further constructions. Doubling the cube requires us to be able to solve the equation x3 − 2 = 0 and trisecting the angle, if it were possible, would enable us to trisect 60◦ (which is con- structible), to obtain 20◦. -
Construction of Regular Polygons a Constructible Regular Polygon Is One That Can Be Constructed with Compass and (Unmarked) Straightedge
DynamicsOfPolygons.org Construction of regular polygons A constructible regular polygon is one that can be constructed with compass and (unmarked) straightedge. For example the construction on the right below consists of two circles of equal radii. The center of the second circle at B is chosen to lie anywhere on the first circle, so the triangle ABC is equilateral – and hence equiangular. Compass and straightedge constructions date back to Euclid of Alexandria who was born in about 300 B.C. The Greeks developed methods for constructing the regular triangle, square and pentagon, but these were the only „prime‟ regular polygons that they could construct. They also knew how to double the sides of a given polygon or combine two polygons together – as long as the sides were relatively prime, so a regular pentagon could be drawn together with a regular triangle to get a regular 15-gon. Therefore the polygons they could construct were of the form N = 2m3k5j where m is a nonnegative integer and j and k are either 0 or 1. The constructible regular polygons were 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ... but the only odd polygons in this list are 3,5 and 15. The triangle, pentagon and 15-gon are the only regular polygons with odd sides which the Greeks could construct. If n = p1p2 …pk where the pi are odd primes then n is constructible iff each pi is constructible, so a regular 21-gon can be constructed iff both the triangle and regular 7-gon can be constructed. -
The Art of the Intelligible
JOHN L. BELL Department of Philosophy, University of Western Ontario THE ART OF THE INTELLIGIBLE An Elementary Survey of Mathematics in its Conceptual Development To my dear wife Mimi The purpose of geometry is to draw us away from the sensible and the perishable to the intelligible and eternal. Plutarch TABLE OF CONTENTS FOREWORD page xi ACKNOWLEDGEMENTS xiii CHAPTER 1 NUMERALS AND NOTATION 1 CHAPTER 2 THE MATHEMATICS OF ANCIENT GREECE 9 CHAPTER 3 THE DEVELOPMENT OF THE NUMBER CONCEPT 28 THE THEORY OF NUMBERS Perfect Numbers. Prime Numbers. Sums of Powers. Fermat’s Last Theorem. The Number π . WHAT ARE NUMBERS? CHAPTER 4 THE EVOLUTION OF ALGEBRA, I 53 Greek Algebra Chinese Algebra Hindu Algebra Arabic Algebra Algebra in Europe The Solution of the General Equation of Degrees 3 and 4 The Algebraic Insolubility of the General Equation of Degree Greater than 4 Early Abstract Algebra 70 CHAPTER 5 THE EVOLUTION OF ALGEBRA, II 72 Hamilton and Quaternions. Grassmann’s “Calculus of Extension”. Finite Dimensional Linear Algebras. Matrices. Lie Algebras. CHAPTER 6 THE EVOLUTION OF ALGEBRA, III 89 Algebraic Numbers and Ideals. ABSTRACT ALGEBRA Groups. Rings and Fields. Ordered Sets. Lattices and Boolean Algebras. Category Theory. CHAPTER 7 THE DEVELOPMENT OF GEOMETRY, I 111 COORDINATE/ALGEBRAIC/ANALYTIC GEOMETRY Algebraic Curves. Cubic Curves. Geometric Construction Problems. Higher Dimensional Spaces. NONEUCLIDEAN GEOMETRY CHAPTER 8 THE DEVELOPMENT OF GEOMETRY, II 129 PROJECTIVE GEOMETRY DIFFERENTIAL GEOMETRY The Theory of Surfaces. Riemann’s Conception of Geometry. TOPOLOGY Combinatorial Topology. Point-set topology. CHAPTER 9 THE CALCULUS AND MATHEMATICAL ANALYSIS 151 THE ORIGINS AND BASIC NOTIONS OF THE CALCULUS MATHEMATICAL ANALYSIS Infinite Series. -
Advanced Mathematics for Secondary Teachers: a Capstone Experience
Advanced Mathematics for Secondary Teachers: A Capstone Experience Curtis Bennett David Meel c 2000 January 3, 2001 2 Contents 1 Introduction 5 1.1 A Description of the Course . 5 1.2 What is Mathematics? . 6 1.3 Background . 9 1.3.1 GCDs and the Fundamental Theorem of Arithmetic . 10 1.3.2 Abstract Algebra and polynomials . 11 1.4 Problems . 14 2 Rational and Irrational Numbers 15 2.1 Decimal Representations . 18 2.2 Irrationality Proofs . 23 2.3 Irrationality of e and π ...................... 26 2.4 Problems . 30 3 Constructible Numbers 35 3.1 The Number Line . 37 3.2 Construction of Products and Sums . 39 3.3 Number Fields and Vector Spaces . 44 3.4 Impossibility Theorems . 54 3.5 Regular n-gons . 58 3.6 Problems . 60 4 Solving Equations by Radicals 63 4.1 Solving Simple Cubic Equations . 66 4.2 The General Cubic Equation . 72 4.3 The Complex Plane . 74 4.4 Algebraic Numbers . 78 4.5 Transcendental Numbers . 80 3 4 CONTENTS 5 Dedekind Cuts 87 5.1 Axioms for the Real Numbers . 87 5.2 Dedekind Cuts . 91 6 Classical Numbers 105 6.1 The Logarithmic Function . 105 7 Cardinality Questions 109 8 Finite Difference Methods 111 Chapter 1 Introduction 1.1 A Description of the Course This course is intended as a capstone experience for mathematics students planning on becoming high school teachers. This is not to say that this is a course in material from the high school curriculum. Nor is this a course on how to teach topics from the high school curriculum. -
FROM GEOMETRY to NUMBER the Arithmetic Field Implicit in Geometry
FROM GEOMETRY TO NUMBER The Arithmetic Field implicit in Geometry Johan Alm Contents Preface v Chapter 1. Kinds of Number in Early Geometry 1 1. Pythagorean natural philosophy 1 2. Plato and Euclid 4 Chapter 2. Fields and Geometric Construction 7 1. Line segment arithmetic 7 2. The Hilbert field 12 3. The constructible field 13 4. The real field 15 Chapter 3. The main theorems 17 1. The Cartesian space over a field 17 2. Faithful models 17 3. Three theorems 18 4. Proof of the first theorem 19 5. Proof of the second theorem 22 6. Proof of the third theorem 26 7. The main theorem 26 Chapter 4. Some applications 29 1. Duplicating the cube 29 2. Trisecting the angle 32 3. Squaring the circle 33 Appendix A. Hilbert’s axioms 35 Appendix B. Axioms and properties of groups and fields 39 Appendix C. Field extensions 41 Appendix. Bibliography 43 iii Preface This is a short monograph on geometry, written by the author as a supplement to David C. Kay’s ”College Geometry” while taking a course based on that book. The presentation in this monograph is very indebted to Robin Hartshorne’s book ”Geometry: Euclid and Beyond.” To be more precise we might say this is a text on classical, synthetic, or ax- iomatic geometry, but we consider this the heart of geometry and prefer to refer to the subject simply as ”geometry.” Most texts on geometry focus on the parallel axiom. Little fuzz is made about the other axioms, and one easily gets the idea that they serve only to provide a sensible foundation. -
Ruler and Compass Constructions and Abstract Algebra
Ruler and Compass Constructions and Abstract Algebra Introduction Around 300 BC, Euclid wrote a series of 13 books on geometry and number theory. These books are collectively called the Elements and are some of the most famous books ever written about any subject. In the Elements, Euclid described several “ruler and compass” constructions. By ruler, we mean a straightedge with no marks at all (so it does not look like the rulers with centimeters or inches that you get at the store). The ruler allows you to draw the (unique) line between two (distinct) given points. The compass allows you to draw a circle with a given point as its center and with radius equal to the distance between two given points. But there are three famous constructions that the Greeks could not perform using ruler and compass: • Doubling the cube: constructing a cube having twice the volume of a given cube. • Trisecting the angle: constructing an angle 1/3 the measure of a given angle. • Squaring the circle: constructing a square with area equal to that of a given circle. The Greeks were able to construct several regular polygons, but another famous problem was also beyond their reach: • Determine which regular polygons are constructible with ruler and compass. These famous problems were open (unsolved) for 2000 years! Thanks to the modern tools of abstract algebra, we now know the solutions: • It is impossible to double the cube, trisect the angle, or square the circle using only ruler (straightedge) and compass. • We also know precisely which regular polygons can be constructed and which ones cannot. -
Constructible Numbers Weeks 7-8 UCSB 2014
Math/CS 120: Intro. to Math Professor: Padraic Bartlett Lecture 6: Constructible Numbers Weeks 7-8 UCSB 2014 As the geometer who sets himself To square the circle and who cannot find, For all his thought, the principle he needs, Just so was I on seeing this new vision. Dante Alighieri, Paradise, canto XXXIII 1 Constructible Numbers Construction, in a sense, was the unofficial theme of the first five weeks of this class. We started our lectures by first defining how to make proofs; from there, we used our language of \proof" to construct N; Z; Q; R and C. However, the method in which we constructed these objects was remarkably abstract; we started with nothing,1 and in the words of J´anosBolyai, \Out of nothing [we] have created a strange new universe." While this was beautiful in its simplicity | given absolutely nothing but a small handful of logical axioms, we were able to build our way to the foundations of modern mathematics in a few weeks | it also makes one wonder if there are perhaps different or easier ways to approach the same constructions. That is: when we conceive of the number 4, we usually don't think of f;; f;g; f;; f;gg; f;; f;g; f;; f;gggg: These notes are meant to offer a different construction for many of the numbers we care about: the constructible numbers! We define them as follows: 2 Definition. Suppose that you are given an infinite piece of paper (think of this as R ,) and two tools: 1.A compass. -
On Constructible Numbers and Angles Original Notes Adopted from March 5, 2002 (Week 21)
More on Constructible Numbers and Angles Original Notes adopted from March 5, 2002 (Week 21) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Lemma: If x0 is a root of a polynomial with coefficients in F(r), then x0 is a root of a polynomial with coefficients in F. n n-1 Proof: Given (an + bn √r) x0 + (an-1 + bn-1 √r) x0 + ... +(a1 + b1 √r)x0 + a0 + b0 √r =0 with ai , bi ∈ F. n By Dividing through by an + bn √r, we can assume it is poly monic. (ie. coefficients of x0 is 1). n n-1 x0 (an-1 + bn-1 √r) x0 + ... +(a1 + b1 √r)x0 + a0 + b0 √r =0 Then, n n-1 n-1 x0 + an-1 x0 +....+ a1x0 + a0 = √r (bn-1x0 + ... + b1x0 + b0 ) Square both Sides n n-1 2 n-1 2 (x0 + an-1 x0 +....+ a1x0 + a0 ) - r (bn-1x0 + ... + b1x0 + b0 ) =0 Recall: x0 algebraic if p(x0)= 0 some polynomial p with rational coefficients. Theorem: Every constructible number is algebraic. Proof: Suppose x0 is constructible. There exists Q = F0 ⊂ F1 ⊂ F2 ⊂......⊂ Fk with Fi = Fi-1 (√ri ) some ri ∈ Fi-1 , ri = 0, √ri ∉ Fi-1 such that x0 ∈ Fk Fk = Fk-1 ( ¥ rk ) x0 = a + b ¥ rk, a, b ∈ Fk x0 root of x – ( a + b ¥U k ) = 0. Lemma ⇒ x0 root of polynomial with coefficients in Fk-1 ⇒ x0 root of polynomial with coefficients in Fk-2 ∴ x0 root of polynomial with coefficients in Q. Corollary: The set of constructible numbers is countable. -
The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction) Kimuya .M
I S S N 2 3 4 7 - 1921 Volume 13 Number 4 Journal of Advance in Mathematics The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction) Kimuya .M. Alex1, Josephine Mutembei2 1Meru University of Science and Technology, Kenya, Department of Physical Sciences, Faculty of Physics 2Meru University of Science and Technology, Kenya, Department of Mathematics 1Cell; +254 704600418 2Cell; +254 721567967 [email protected] [email protected] ABSTRACT This paper is devoted to exposition of a provable classical solution for the ancient Greek’s classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837), presented an algebraic proof based on ideas from Galois theory showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the ages old problem of trisecting an arbitrary angle using the traditional Greek’s tools of geometry (classical compass and straightedge), and refute the presented proof of angle trisection impossibility statement. The exposed proof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archim edes proposition of using a marked straightedge construction [4], [11]. Key Words Angle trisection; An arbitrary angle; A particular angle; Compass; Ruler (Straightedge); Classical geometry; GeoGebra Software; Plane geometry; Subset; Superset; Singular angle Notations Notation for an angle Denotes a straight line segment and a length Two Dimensions Three Dimensions Four significant figures Denotes a member or an element of a particular set Subset Not a member or an element of a particular set divides Cosine of angle decimal places Academic Discipline And Sub-Disciplines Mathematics (Geometry) 1.0 INTRODUCTION Compass and straightedge problems have always been the favorite subject o f classical geometry. -
Transcendental Numbers
Bachelor Thesis DEGREE IN MATHEMATICS Faculty of Mathematics University of Barcelona TRANSCENDENTAL NUMBERS Author: Adina Nedelea Director: Dr. Ricardo Garc´ıaL´opez Department: Algebra and Geometry Barcelona, June 27, 2016 Summary Transcendental numbers are a relatively recent finding in mathematics and they provide, togheter with the algebraic numbers, a classification of complex numbers. In the present work the aim is to characterize these numbers in order to see the way from they differ the algebraic ones. Going back to ancient times we will observe how, since the earliest history mathematicians worked with transcendental numbers even if they were not aware of it at that time. Finally we will describe some of the consequences and new horizons of mathematics since the apparition of the transcendental numbers. Agradecimientos El trabajo final de grado es un punto final de una etapa y me gustar´aagradecer a todas las personas que me han dado apoyo durante estos a~nosde estudios y hicieron posible que llegue hasta aqu´ı. Empezar´epor agradecer a quien hizo posible este trabajo, a mi tutor Ricardo Garc´ıa ya que sin su paciencia, acompa~namiento y ayuda me hubiera sido mucho mas dif´ıcil llevar a cabo esta tarea. Sigo con los agradecimientos hacia mi familia, sobretodo hacia mi madre quien me ofreci´ola libertad y el sustento aunque no le fuera f´acily a mi "familia adoptiva" Carles, Marilena y Ernest, quienes me ayudaron a sostener y pasar por todo el pro- ceso de maduraci´onque conlleva la carrera y no rendirme en los momentos dif´ıciles. Gracias por la paciencia y el amor de todos ellos. -
Constructible Numbers Definition 13.1 (A) a Complex Number R Is Algebraic If P(R) = 0 for Some P(X) ∈ Q[X]
Math 4030-001/Foundations of Algebra/Fall 2017 Ruler/Compass at the Foundations: Constructible Numbers Definition 13.1 (a) A complex number r is algebraic if p(r) = 0 for some p(x) 2 Q[x]. Otherwise r is transcendental. (b) Let A ⊂ C be the set of algebraic numbers. (c) Given r 2 A, there is a unique prime polynomial of the form: d d−1 f(x) = x + cd−1x + ··· + c0 2 Q[x] with r as a root. This is the minimal polynomial of r. p n 2π Examples. (a) All rational numbers, i; 2; cos( n ) are algebraic. (b) π; e are transcendental (this isn't easy to prove!). Remark. If r 2 A and v 2 Q[r], then we saw in the previous section that v 2 A, since the vectors 1; v; v2; :::: 2 Q[r] are linearly dependent. Proposition 13.2. The set of algebraic numbers A ⊂ C is a field. Proof. Let r 2 A. Then −r; 1=r 2 Q[r] are also algebraic by the previous remark (or else one can easily find their minimal polynomials). Let s 2 A be another algebraic number. It suffices to show that both r + s and r · s are algebraic numbers to conclude that A is a field. To do this, we make a tower of number fields. Let F = Q[r] and consider F [x], the commutative ring of polynomials with coefficients in F . One such polynomial is the minimal polynomial f(x) 2 Q[x] of s 2 A. This may not be prime in F [x], but it will have a prime factor g(x) with g(s) = 0.