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Proof. Let f and g be nonzero polynomials, and define 9.12 Polynomials over a Field 329 S ={g − fq : q ∈ Q[t]} (9.57) If the zero polynomial is in S, then we have g = fq. Otherwise we may let r be any polynomial in S whose degree is minimal. Then r = g − fq for some polynomial q, so that g = fq + r. To show deg r
330 Chapter 9 Rings the polynomials of degree zero. Thus two polynomials are associates if and only if one is a nonzero constant multiple of the other. Example 9.12.4 Two factorizations of t3 + 3t2 + 2t are t(t + 2)(t + 1) and (4t)( 1 3t + 2 3 )( 3 4t + 3 4 ). Associate pairs are t and 4t, t + 2 and 1 3t + 2 3 , and t + 1 and 3 4t + 3 4 . � We have said a lot about irreducible polynomials in Q[t], but we have not developed criteria by which we can determine whether a polynomial is reducible or irreducible. By Exercise 9.8.22, if deg f = 1 and f = gh, then either g or h has degree zero and is therefore a unit. Thus polynomials of degree one are irreducible. Another simple criterion for irreducibility involves evaluating the polynomial at some rational number. In Section 9.4, we said we were not really interested in polynomials as functions where we would plug in values for t, but more as a string of symbols. Well, that statement is now false. As a ring in and of itself, a polynomial ring is exactly as we described it in Section 9.4, and t is indeed just a formal (and foreign) symbol whose presence is used to define the addition and multiplication of two elements. But for a given polynomial f there are some pretty good reasons we might want to choose a specific ring element α and calculate the value of f(α). The ring element f(α) can make an important statement about the polynomial f and its role in the polynomial ring. You spent time in high school algebra trying to factor polynomials, and one way you might have stumbled onto a linear factor of f was by discovering some number a such that f(a) = 0. For example, if f = t 3 − t 2 + 2t − 2, you might have noticed that f(1) = 0 by adding the coefficients, so you could write f = (t − 1)g, which by division became f = (t − 1)(t 2 + 2). The reason this worked was the following exercise. EXERCISE 9.12.5 Suppose f is a polynomial in Q[t] and a is a rational number. Then (t − a) | f if and only if f(a) = 0. Exercise 9.12.5 makes the following almost immediate. Theorem 9.12.6 Suppose f is a polynomial in Q[t] whose degree is either 2or3.Thenf is reducible if and only if there exists a rational number a such that f(a) = 0. Proof. Let f be a polynomial in Q[t] of degree 2 or 3. (⇒) Suppose f is reducible. Then f = gh, where the degrees of g and h are nonzero, but strictly less than deg f . Without loss of generality, we may assume deg g = 1, so that g = at + b for rational numbers a and b, where a is not zero. Thus f(−b/a) = g(−b/a)h(−b/a) = 0. (⇐) Suppose there exists rational a such that f(a) = 0. By Exercise 9.12.5, (t − a) | f , so that f is reducible.
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
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5.6 Compactness 181 EXERCISE 5.6.3
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5.6 Compactness 183 EXERCISE 5.6.13
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Sequences of Real Numbers 6.1 Seque
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6.1 Sequences Defined 187 Example 6
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EXERCISE 6.1.15 Consider the sequen
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L 1 e L L 2 e . . . . . . 1 2 3 4 N
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6.2 Convergence of Sequences 193 EX
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6.2 Convergence of Sequences 195 To
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6.3 The Nested Interval Property 19
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a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
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Example 6.4.7 The terms a0 = 1 a1 =
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Functions of a Real Variable 7 Func
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7.1 Bounded and Monotone Functions
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50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
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7.2 Limits and Their Basic Properti
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7.2 Limits and Their Basic Properti
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7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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7.6 Implications of Continuity 231
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7.6 Implications of Continuity 233
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7.7 Uniform Continuity 235 we decla
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7.7 Uniform Continuity 237 Next, le
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7.7.2 Uniform Continuity and Compac
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P A R T I I I Basic Principles of A
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Groups 8 In its simplest terms, alg
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8.1 Introduction to Groups 245 are
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8.1 Introduction to Groups 247 Defi
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◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
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Show that C × is an abelian group
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8.2 Subgroups 253 G1 and G3 are aut
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8.2 Subgroups 255 Notice how proper
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8.2 Subgroups 257 Example 8.2.15 su
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EXERCISE 8.2.23 If G is a group and
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8.3 Quotient Groups 261 [0] ={...,
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8.3 Quotient Groups 263 is standard
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Step 2: Define an operation ∗H on
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(iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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then for a given f ∈ S and any a
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Thus f 2 = f ◦ f = (132)(56) ◦
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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Page 322 and 323: 9.3 Ring Properties 299 EXERCISE 9.
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Page 338 and 339: 9.8 Integral Domains 315 EXERCISE 9
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Page 350 and 351: 9.11 Euclidean Domains 327 Exercise
Page 354 and 355: 9.12 Polynomials over a Field 331 W
Page 356 and 357: 9.13 Polynomials over the Integers
Page 358 and 359: 9.14 Ring Morphisms 335 Example 9.1
Page 360 and 361: 9.14 Ring Morphisms 337 EXERCISE 9.
Page 362 and 363: 9.15 Quotient Rings 339 this, howev
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Page 366 and 367: 9.15 Quotient Rings 343 However, th
Page 368 and 369: Index =,51 ɛ (epsilon), 167-168 ɛ
Page 370 and 371: Index 347 Cosets, 263, 264, 265 Lag
Page 372 and 373: Index 349 Gauchy sequences, 202-206
Page 374 and 375: Index 351 identity, 124 relations v
Page 376 and 377: Index 353 Quaternions, 248, 253, 25
Page 378 and 379: Index 355 Tower of Hanoi, 84 Transc
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