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Concrete Algebra With a View Toward Abstract Algebra PDF

204 Pages·2016·2.89 MB·English
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Benjamin McKay Concrete Algebra With a View Toward Abstract Algebra May 16, 2016 ThisworkislicensedunderaCreativeCommonsAttribution-ShareAlike3.0UnportedLicense. iii Preface WithmyfullphilosophicalrucksackIcanonlyclimbslowlyupthemoun- tain of mathematics. — Ludwig Wittgenstein Culture and Value These notes are from lectures given in 2015 at University College Cork. They aim to explain the most concrete and fundamental aspects of algebra, in particular the algebra of the integers and of polynomial functions of a single variable, grounded by proofs using mathematical induction. It is impossible to learn mathematics by reading a book like you would read a novel; you have to work through exercises and calculate out examples. You should try all of the problems. More importantly, since the purpose of this class is to give you a deeper feeling for elementary mathematics, rather than rushing into advanced mathematics, you should reflect about how the following simple ideas reshape your vision of algebra. Consider how you can use your new perspective on elementary mathematics to help you some day guide other students, especially children, with surer footing than the teachers who guided you. v vi The temperature of Heaven can be rather accurately computed. Our authority is Isaiah 30:26, “Moreover, the light of the Moon shall be as the light of the Sun and the light of the Sun shall be sevenfold, as the light of seven days.” Thus Heaven receives from the Moon as much radiation as we do from the Sun, and in addition 7×7 = 49 times as much as the Earth does from the Sun, or 50 times in all. The light we receive from the Moon is one1/10000ofthelightwereceivefromtheSun,sowecanignore that.... The radiation falling on Heaven will heat it to the point wheretheheatlostbyradiationisjustequaltotheheatreceivedby radiation,i.e.,Heavenloses50timesasmuchheatastheEarthby radiation. Using the Stefan-Boltzmann law for radiation, (H/E) temperature of the earth (∼ 300K), gives H as 798K (525◦C). The exact temperature of Hell cannot be computed.... [However] Revelations 21:8 says “But the fearful, and unbelieving ...shall havetheirpartinthelakewhichburnethwithfireandbrimstone.” Alakeofmolten brimstonemeansthatitstemperaturemustbeat or below the boiling point, 444.6◦C. We have, then, that Heaven, at 525◦C is hotter than Hell at 445◦C. — Applied Optics , vol. 11, A14, 1972 Inthesedaystheangeloftopologyandthedevilofabstractalgebra fight for the soul of every individual discipline of mathematics. — Hermann Weyl Invariants,DukeMathematicalJournal5,1939,489– 502 — and so who are you, after all? —Iampartofthepowerwhichforeverwillsevilandforeverworks good. — Goethe Faust This Book is not to be doubted. — Quran , 2:1/2:6-2:10 The Cow Contents 1 The integers 1 2 Mathematical induction 9 3 Greatest common divisors 15 4 Prime numbers 21 5 Modular arithmetic 25 6 Secret messages 39 7 Rational, real and complex numbers 45 8 Polynomials 53 9 Factoring polynomials 59 10 Resultants and discriminants 69 11 Permuting roots 79 12 Fields 91 13 Rings 101 14 Algebraic curves in the plane 107 15 Quotient rings 121 16 Field extensions and algebraic curves 127 17 The projective plane 129 18 Algebraic curves in the projective plane 135 19 Families of plane curves 147 20 The tangent line 155 21 Projective duality 165 22 Polynomial equations have solutions 167 23 More projective planes 171 Bibliography 189 List of notation 191 Index 193 vii Chapter 1 The integers God made the integers; all else is the work of man. — Leopold Kronecker Notation We will write numbers using notation like 1234567.12345, using a decimal point . at the last integer digit, and using thin spaces to separate out every 3 digits before or afterthedecimalpoint. Youmightprefer1,234,567·123,45or1,234,567.123,45,which arealsofine. Wereservethe·symbolformultiplication, writing2·3=6ratherthan 2×3=6. The laws of integer arithmetic The integers are the numbers ...,−2,−1,0,1,2,.... Let us distill their essential properties, using only the concepts of addition and multiplication. Addition laws: a. The associative law: For any integers a,b,c: (a+b)+c=a+(b+c). b. Theidentitylaw:Thereisaninteger0sothatforanyintegera,a+0=a. c. Theexistenceofnegatives:foranyintegera,thereisanintegerb(denote by the symbol −a) so that a+b=0. d. The commutative law: For any integers a,b, a+b=b+a. Multiplication laws: a. The associative law: For any integers a,b,c: (ab)c=a(bc). b. The identity law: There is an integer 1 so that for any integer a, a1=a. c. The zero divisors law: For any integers a,b, if ab=0 then a=0 or b=0. d. The commutative law: For any integers a,b, ab=ba. The distributive law: a. For any integers a,b,c: a(b+c)=ab+ac. 1 2 The integers Sign laws: Anintegeraispositiveifitliesamongtheintegers1,1+1,1+1+1,...,negative if −a is positive. Of course, we write 1+1 as 2, and 1+1+1 as 3 and so on. We write a < b to mean that there is a positive number c so that b = a+c, write a>b to mean b<a, write a≤b to mean a<b or a=b, and so on. We write|a|tomeana,ifa≥0,andtomean−aotherwise,andcallittheabsolute value of a. a. The inequality cancellation law for addition: For any integers a,b,c, if a+c<b+c then a<b. b. The inequality cancellation law for multiplication: If a < b and if c > 0 then ac<bc. c. Determinacy of sign: Every integer a has precisely one of the following properties: a>0, a<0, a=0. The law of well ordering: a. Any collection of positive integers has a least element; that is to say, an element a so that every element b satisfies a≤b. All of the other arithmetic laws we are familiar with can be derived from these. For example,theassociativelawforaddition,appliedtwice,showsthat(a+b)+(c+d)= a+(b+(c+d)), and so on, so that we can add up any finite sequence of integers, in any order, and get the same result, which we write in this case as a+b+c+d. A similar story holds for multiplication. To understand mathematics, you have to solve a large number of problems. I prayed for twenty years but received no answer until I prayed with my legs. — Frederick Douglass, statesman and escaped slave 1.1 For each equation below, what law above justifies it? a. 7(3+1)=7·3+7·1 b. 4(9·2)=(4·9)2 c. 2·3=3·2 1.2 Usethelawsabovetoprovethatifa,b,careanyintegersthen(a+b)c=ac+bc. 1.3 Use the laws above to prove that 0+0=0. 1.4 Use the laws above to prove that 0·0=0. 1.5 Use the laws above to prove that, for any integer a, a·0=0. 1.6 Use the laws above to prove that (−1)(−1)=1. 1.7 Use the laws above to prove that the product of any two negative numbers is positive.

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