List of results

Definition 6.4.1 Evaluation, Root, Polynomial Function
Theorem 6.4.1 Evaluation Respects the Operations
Definition 6.4.2 Ring Homomorphism
Example Ring Homomorphisms and Two Impostors
Theorem 6.4.2 Substitution Principle
Corollary 6.4.3 Changing the Coefficient Ring
Definition 6.4.3 Composition of Polynomials
Proposition 6.4.4 Polynomials and Their Functions
Example Different Polynomials, Equal Functions
Exercise A1
Exercise B1: The degree of a composition
Exercise B2: All polynomials of degree at most two over the field with two elements
Exercise B3: Shifting the variable
Exercise C1: Exit check: the substitution principle
Exercise C2: Composition is associative
Exercise C3: The factor theorem over any commutative ring
Exercise C4: A second proof of Fermat’s little theorem

6.5 The Formal Derivative

Definition 6.5.1 Formal Derivative
Example Formal derivatives in five rings
Theorem 6.5.1 Rules for the Formal Derivative
Proposition 6.5.2 When the Formal Derivative Is Zero
Exercise A1
Exercise B1: The chain rule
Exercise B2: Higher derivatives
Exercise C1: Derivations
Exercise C2: Find the error
Exercise R1
Exercise R2
Exercise R3
Exercise R4

Polynomials over a Field

7.1 Division with Remainder

Definition 7.1.1 Divisibility of Polynomials
Proposition 7.1.1 Rules for Divisibility of Polynomials
Theorem 7.1.2 Division with Remainder in $F[x]$
Example Long division in two fields
Theorem 7.1.3 Remainder Theorem and Factor Theorem
Example Remainders without dividing
Exercise A1
Exercise B1: Two long divisions
Exercise B2: A factor of a difference of powers
Exercise B3: One remainder over every field
Exercise C1: Monic divisors keep integer coefficients
Exercise C2: The remainder modulo a quadratic with known roots

7.2 Ideals, the Greatest Common Divisor, and the Euclidean Algorithm

Definition 7.2.1 Ideal Generated by Polynomials
Theorem 7.2.1 Ideals of $F[x]$ Are Principal
Definition 7.2.2 Greatest Common Divisor of Polynomials
Theorem 7.2.2 Bézout’s Theorem for Polynomials
Definition 7.2.3 Coprime Polynomials
Corollary 7.2.3 Coprime Means 1 Is a Combination
Lemma 7.2.4 One Step of the Euclidean Algorithm
Theorem 7.2.5 The Euclidean Algorithm for Polynomials
Example A run of the Euclidean algorithm
Proposition 7.2.6 Division and gcd Do Not Depend on the Field
Exercise A1
Exercise B1: A gcd and its common root
Exercise B2: A gcd over the field with two elements
Exercise C1: Exit check: ideals of $F[x]$ and Bézout
Exercise C2: Scaling the gcd
Exercise C3: The gcd of two binomials

7.3 Irreducible Polynomials and Unique Factorization

Definition 7.3.1 Irreducible Polynomial
Proposition 7.3.1 Irreducibility in Low Degree
Theorem 7.3.2 Euclid’s Lemma for Polynomials
Corollary 7.3.3 An Irreducible Dividing a Product
Proposition 7.3.4 Coprime Divisors of a Polynomial
Lemma 7.3.5 An Irreducible Factor Exists
Theorem 7.3.6 Unique Factorization of Polynomials
Example The Factorizations of $x^4 - 1$
Theorem 7.3.7 Rational Root Test
Example A Cubic with No Rational Root
Exercise A1
Exercise B1: Determine which
Exercise B2: The polynomials vanishing at the square root of two
Exercise B3: The small irreducibles over the two-element field
Exercise C1: Exit check: unique factorization
Exercise C2: Find the error
Exercise C3: Infinitely many irreducibles
Exercise C4: Greatest common divisor and least common multiple

7.4 Roots, Multiplicity, and the Formal Derivative

Theorem 7.4.1 Counting Roots
Theorem 7.4.2 Polynomials Determined by Values
Example Polynomial Functions over a Finite Field
Theorem 7.4.3 Expansion in Powers of $x - c$
Definition 7.4.1 Multiplicity of a Root
Lemma 7.4.4 Multiplicity through the Cofactor
Proposition 7.4.5 Multiplicity of a Product
Theorem 7.4.6 Splitting Off All Roots
Definition 7.4.2 Splitting over a Field
Example Multiplicities by Repeated Division
Theorem 7.4.7 Repeated Roots and the Derivative
Corollary 7.4.8 Repeated Factors and the Greatest Common Divisor
Exercise A1
Exercise B1: Put the proof in order
Exercise B2: Expansions at a point
Exercise B3: Functions on a finite field
Exercise C1: The coefficients as derivatives
Exercise C2: Repeated factors through the gcd

7.5 Interpolation, Quotient Rings, and the Chinese Remainder Theorem

Definition 7.5.1 Lagrange Polynomials
Theorem 7.5.1 Lagrange Interpolation
Example A cubic through four points
Definition 7.5.2 Congruence Modulo a Polynomial
Theorem 7.5.2 The Ring of Polynomials Modulo m
Theorem 7.5.3 The Quotient Is a Field Exactly When m Is Irreducible
Example A field with four elements, and the complex numbers again
Lemma 7.5.4 Coprime to Each Factor, Coprime to the Product
Theorem 7.5.5 Chinese Remainder Theorem for Polynomials
Proposition 7.5.6 Interpolation Is the Chinese Remainder Theorem for Linear Moduli
Example A linear and a quadratic modulus
Example Values and derivatives at once
Exercise A1
Exercise B1
Exercise B2
Exercise B3
Exercise C1: Two routes to the same theorem
Exercise C2: Every function on a prime field is a polynomial

7.6 The Fundamental Theorem of Algebra

Lemma 7.6.1 Growth of a Complex Polynomial
Lemma 7.6.2 A Lipschitz Bound for a Complex Polynomial
Theorem 7.6.3 The Modulus of a Complex Polynomial Attains Its Minimum
Lemma 7.6.4 A Direction with Negative Real Part
Lemma 7.6.5 d’Alembert’s Lemma
Example One step of descent
Theorem 7.6.6 Fundamental Theorem of Algebra
Corollary 7.6.7 Complex Polynomials Split
Example Factoring a quartic over the reals and over the complex numbers
Proposition 7.6.8 Conjugate Roots of Real Polynomials
Corollary 7.6.9 The Irreducible Real Polynomials
Theorem 7.6.10 Factorization of Real Polynomials
Corollary 7.6.11 Real Polynomials of Odd Degree Have a Real Root
Exercise A1
Exercise B1
Exercise B2
Exercise B3
Exercise C1: Exit check: the descent and the theorem
Exercise C2: Faded: the real polynomials vanishing at a non-real number
Exercise C3: Find the error
Exercise R1
Exercise R2
Exercise R3
Exercise R4

Real Functions and Their Graphs

8.1 Real Functions and Their Graphs

Definition 8.1.1 Real Function, Graph
Example First Real Functions
Proposition 8.1.1 Vertical Line Test
Definition 8.1.2 Operations on Real Functions
Definition 8.1.3 Polynomial Function, Rational Function
Example Method: reading a graph as a set of pairs
Proposition 8.1.2 Moving a Graph
Definition 8.1.4 Even and Odd Functions
Proposition 8.1.3 Even and Odd Parts
Exercise A1
Exercise A2: Is this legal? Is it useful?
Exercise B1
Exercise B2
Exercise B3
Exercise B4: Prove or disprove
Exercise C1: Additive functions, used later

8.2 Monotone Functions and Inverse Functions

Definition 8.2.1 Monotone Function
Example Method: factor out y − x
Example A quotient of two increasing functions
Example The squashing function
Proposition 8.2.1 Rules for Monotone Functions
Proposition 8.2.2 Strictly Monotone Functions Reflect the Order
Example Injective but not monotone
Theorem 8.2.3 Inverse of a Strictly Monotone Function
Example Roots as inverse functions
Example A strictly increasing function with a jump
Exercise A1
Exercise A2
Exercise B1
Exercise B2: Prove or disprove
Exercise B3
Exercise C1: Find the error
Exercise C2, used later

8.3 Bounded Functions and Their Suprema

Proposition 8.3.1 Infimum by Reflection
Proposition 8.3.2 The Epsilon Test for a Supremum
Definition 8.3.1 Bounded Function
Definition 8.3.2 Supremum and Infimum of a Function
Example Method: bound every value, then find a witness
Example A supremum that is not attained
Example A supremum that is attained
Theorem 8.3.3 Rules for Suprema of Functions
Proposition 8.3.4 The Spread of Values
Example Polynomials of positive degree are unbounded
Proposition 8.3.5 Iterated Suprema
Exercise A1
Exercise B1
Exercise B2: Prove or disprove
Exercise B3
Exercise B4: Find the error
Exercise C1: Chapter check: the supremum of a square
Exercise C2

8.4 Floor, Distance to the Integers, and Step Functions

Definition 8.4.1 Floor, Ceiling, Fractional Part
Example Method: pin the floor by its inequality
Proposition 8.4.1 Rules for the Floor
Definition 8.4.2 Distance to the Integers
Proposition 8.4.2 Distance to the Integers
Definition 8.4.3 Periodic Function
Definition 8.4.4 Step Function
Proposition 8.4.3 Step Functions Are Closed Under the Operations
Exercise A1
Exercise A2
Exercise B1: Prove or disprove
Exercise B2: Counting multiples
Exercise B3
Exercise C1: A sum of floors
Exercise C2: Multiplying the distance

8.5 The Exponential, the Logarithm, and Real Powers

Axiom 8.5.1 What This Volume Borrows about the Exponential
Proposition 8.5.2 First Consequences of the Addition Rule and the Tangent-Line Bound
Example Method: squeeze, then close the gap by Archimedes
Theorem 8.5.3 The Exponential Is Determined by Its Addition Rule and Tangent-Line Bound
Definition 8.5.1 Euler’s Number
Definition 8.5.2 The Logarithm
Proposition 8.5.4 Rules for the Logarithm
Definition 8.5.3 Real Powers
Proposition 8.5.5 Real Powers Agree with Roots and Integer Powers
Proposition 8.5.6 Laws of Real Powers
Proposition 8.5.7 The Exponential Beats Every Power
Exercise A1
Exercise A2
Exercise B1
Exercise B2: Prove or disprove
Exercise B3: Put the proof in order
Exercise C1: Find the error
Exercise C2: Chapter check: the logarithm is slower than every root