List of results

Theorems and lemmas 118 Definitions 37 Examples 46 Exercises 195

Complex Numbers

1.1 The Complex Numbers as Pairs of Reals

Definition 1.1.1 Complex Numbers
Proposition 1.1.1 The Real Numbers inside the Complex Numbers
Example First Computations in $\mathbb{C}$
Proposition 1.1.2 Arithmetic Laws in $\mathbb{C}$
Proposition 1.1.3 Laws of Powers in a Commutative Ring
Lemma 1.1.4 Difference of Powers
Exercise A1
Exercise B1: Powers of $i$
Exercise B2
Exercise B3: The zero divisors of the minimal change
Exercise C1: There is only one way to adjoin a square root of minus one

1.2 Conjugate, Modulus, and Division

Definition 1.2.1 Conjugate and Modulus
Proposition 1.2.1 The Modulus Extends the Absolute Value
Theorem 1.2.2 Properties of Conjugation and Modulus
Proposition 1.2.3 Inverse of a Complex Number
Theorem 1.2.4 The Complex Numbers Form a Field
Example Dividing Complex Numbers
Exercise A1
Exercise B1: Arithmetic
Exercise B2
Exercise B3
Exercise C1: Exit check: the complex numbers form a field
Exercise C2: Gaussian rationals

1.3 The Triangle Inequality

Lemma 1.3.1 Real and Imaginary Parts against the Modulus
Theorem 1.3.2 Triangle Inequality for Complex Numbers
Corollary 1.3.3 Variants of the Triangle Inequality
Exercise A1
Exercise B1: The equality case
Exercise B2: A distance that does not change
Exercise B3
Exercise C1: Exit check: the triangle inequality
Exercise C2: Inside the unit disc

1.4 Square Roots, Quadratic Equations, and the Missing Order

Proposition 1.4.1 Square Roots of Complex Numbers
Example Square Roots of $5 - 12i$
Corollary 1.4.2 Quadratic Formula over the Complex Numbers
Proposition 1.4.3 The Complex Numbers Cannot Be Ordered
Exercise A1
Exercise B1
Exercise B2: Put the proof in order
Exercise B3
Exercise C1: Find the error
Exercise C2: A second proof that there is no order

1.5 The Unit Circle and Polar Form

Proposition 1.5.1 The Unit Circle
Axiom 1.5.2 What This Volume Borrows about Cosine and Sine
Definition 1.5.1 Polar Form
Theorem 1.5.3 Multiplication in Polar Form
Proposition 1.5.4 Existence and Uniqueness of Polar Form
Theorem 1.5.5 De Moivre’s Formula
Example Special Angles
Example A Power via Polar Form
Exercise A1
Exercise B1
Exercise B2: Symmetries of Cosine and Sine
Exercise B3
Exercise C1: Triple Angles and Negative Powers
Exercise C2: Points on the Circle without Angles

1.6 Higher Roots, Roots of Unity, and a Look Ahead

Lemma 1.6.1 Real n-th Roots
Theorem 1.6.2 n-th Roots of a Complex Number
Example Roots of Unity
Proposition 1.6.3 Sums of Roots of Unity
Proposition 1.6.4 Real Coefficients and Conjugate Roots
Exercise A1
Exercise B1
Exercise B2
Exercise B3
Exercise C1: Exit check: n-th roots
Exercise C2: Find the error
Exercise R1
Exercise R2
Exercise R3
Exercise R4

Fields

2.1 The Field Axioms and the First Examples

Definition 2.1.1 Field
Theorem 2.1.1 Basic Properties of Fields
Corollary 2.1.2 Equal Squares
Proposition 2.1.3 Linear Equations in a Field
Example The Fields $\mathbb{Q}$ , $\mathbb{R}$ and $\mathbb{C}$
Example A New Multiplication on $\mathbb{Q}$
Example Four Systems That Are Not Fields
Exercise A1
Exercise B1
Exercise B2
Exercise B3: Which Are Fields?
Exercise C1: Exit check: rules from the axioms
Exercise C2: Find the error

2.2 Finite Fields from Modular Arithmetic

Example Arithmetic on $\mathbb{Z}/n\mathbb{Z}$ and the Field $\mathbb{F}_2$
Lemma 2.2.1 Bézout’s Lemma for Integers
Theorem 2.2.2 $\mathbb{Z}/n\mathbb{Z}$ Is a Field Exactly for Prime $n$
Definition 2.2.1 The Field $\mathbb{F}_p$
Example Inverses in $\mathbb{F}_7$ and in $\mathbb{F}_{101}$
Example Elimination over $\mathbb{F}_5$
Exercise A1
Exercise B1
Exercise B2: Which Are Fields?
Exercise B3
Exercise C1: A Field with Four Elements
Exercise C2: Find the error
Exercise C3

2.3 Subfields

Definition 2.3.1 Subfield
Proposition 2.3.1 The Subfield Test
Example The Field Generated by the Square Root of Two
Example Rational and Integer Complex Numbers
Proposition 2.3.2 Intersections of Subfields
Definition 2.3.2 Prime Subfield
Exercise A1
Exercise B1: Putting a proof in order
Exercise B2
Exercise B3
Exercise B4
Exercise C1: A finite subset closed under the operations
Exercise C2: Find the error

2.4 Field Homomorphisms

Definition 2.4.1 Field Homomorphism
Proposition 2.4.1 Properties of Field Homomorphisms
Example Conjugation and Other Automorphisms
Example Pairs of Fields with No Homomorphism
Exercise A1
Exercise B1: Images of quotients
Exercise B2: The automorphisms of the rational complex numbers
Exercise B3
Exercise B4: The fixed field
Exercise C1: Isomorphic fields have the same square roots of minus one
Exercise C2: Find the error

2.5 Integer Multiples, the Characteristic, and the Prime Subfield

Lemma 2.5.1 Integer Multiples
Definition 2.5.1 Characteristic
Proposition 2.5.2 The Characteristic Divides Every Vanishing Multiple
Example Completing the square in $\mathbb{F}_3$
Theorem 2.5.3 The Characteristic Is Zero or a Prime
Proposition 2.5.4 Subfields and Homomorphisms Preserve the Characteristic
Example The characteristics of the standard fields
Theorem 2.5.5 The Prime Subfield
Corollary 2.5.6 Homomorphisms Respect the Prime Subfield
Corollary 2.5.7 A Finite Field Has Prime Characteristic
Exercise A1
Exercise B1: Reducing multiples
Exercise B2: Completing the square
Exercise B3: When the integers embed
Exercise C1: Exit check: the characteristic
Exercise C2: Find the error
Exercise C3: The four-element field

2.6 The Binomial Theorem, the Frobenius Map, and Fermat's Little Theorem

Lemma 2.6.1 Finite Sums in a Commutative Ring
Definition 2.6.1 Binomial Coefficients
Proposition 2.6.2 Absorption Identity
Theorem 2.6.3 The Binomial Theorem in a Commutative Ring
Lemma 2.6.4 A Prime Divides the Middle Binomial Coefficients
Theorem 2.6.5 The Frobenius Homomorphism
Theorem 2.6.6 Fermat’s Little Theorem
Example Powers and inverses in prime fields
Exercise A1
Exercise B1
Exercise B2
Exercise B3
Exercise C1: Exit check: the Frobenius identity
Exercise C2: Fermat by induction
Exercise C3: Roots of order p
Exercise R1
Exercise R2
Exercise R3
Exercise R4

Groups, Subgroups, and Homomorphisms

3.1 Groups

Definition 3.1.1 Group
Example Groups
Example Not Groups
Theorem 3.1.1 Basic Properties of Groups
Definition 3.1.2 Product of a List
Proposition 3.1.2 Laws for Products of Lists
Definition 3.1.3 Integer Powers
Proposition 3.1.3 Power Laws in a Group
Proposition 3.1.4 The Units Form a Group
Exercise A1
Exercise B1: Groups or not
Exercise B2: An element equal to its square
Exercise B3: Powers of a conjugate
Exercise C1: Exit check: the basic laws of a group
Exercise C2: Find the error
Exercise C3: Rearranging in an abelian group

3.2 Subgroups and the Subgroups of the Integers

Definition 3.2.1 Subgroup
Theorem 3.2.1 Subgroup Test
Example Subgroups
Proposition 3.2.2 Intersections of Subgroups
Definition 3.2.2 Subgroup Generated by an Element
Proposition 3.2.3 The Subgroup Generated by an Element
Theorem 3.2.4 Subgroups of the Integers
Exercise A1
Exercise B1: Put the proof in order
Exercise B2: Subgroups of the integers modulo twelve
Exercise B3: Integer combinations, once more
Exercise C1: Finite subsets closed under products
Exercise C2: Exit check: subgroups of the integers
Exercise C3: Inclusions and intersections

3.3 Homomorphisms and Isomorphisms

Definition 3.3.1 Group Homomorphism
Example Homomorphisms
Theorem 3.3.1 Basic Properties of Homomorphisms
Proposition 3.3.2 Composites and Inverses of Homomorphisms
Proposition 3.3.3 Injective Exactly When the Kernel Is Trivial
Example Isomorphic and Non-Isomorphic Small Groups
Exercise A1
Exercise B1
Exercise B2
Exercise B3
Exercise C1
Exercise C2

3.4 Permutations and Cycle Notation

Definition 3.4.1 Permutations and the Symmetric Group
Definition 3.4.2 Cycle, Transposition
Example Computing with Cycles
Lemma 3.4.1 Permutations Fixing the Last Point
Proposition 3.4.2 The Symmetric Group Has n! Elements
Proposition 3.4.3 Disjoint Permutations Commute
Theorem 3.4.4 Disjoint Cycle Decomposition
Exercise A1
Exercise B1: The table of $S_3$
Exercise B2: Counting cycles
Exercise B3: The rotations of a square
Exercise C1: Dropping “disjoint”
Exercise C2: Powers of disjoint cycles

3.5 Transpositions Generate the Symmetric Group

Proposition 3.5.1 A Cycle Is a Product of Transpositions
Theorem 3.5.2 Transpositions Generate $S_n$
Corollary 3.5.3 Adjacent Transpositions Generate $S_n$
Exercise A1
Exercise B1
Exercise B2
Exercise C1: Exit check: transpositions generate
Exercise C2: Find the error
Exercise C3
Exercise R1
Exercise R2
Exercise R3
Exercise R4

Formal Polynomials

4.1 Polynomials as Coefficient Sequences

Definition 4.1.1 Polynomial
Example Reading off coefficient sequences
Lemma 4.1.1 Finite Sums, Powers and Multiples
Exercise A1
Exercise B1: Polynomials of bounded length as tuples
Exercise B2: Determine which
Exercise C1: Finitely many, in the sense of finite sets

4.2 The Polynomial Ring

Definition 4.2.1 The Polynomial Ring
Example A product, coefficient by coefficient
Proposition 4.2.1 A Polynomial Is the Sum of Its Monomials
Lemma 4.2.2 Multiplying by a Monomial
Lemma 4.2.3 Check on Monomials
Theorem 4.2.4 The Polynomial Ring Is a Commutative Ring
Exercise A1
Exercise B1: A product over two rings
Exercise B2: Changing the sign of x
Exercise C1: Exit check: the ring laws
Exercise C2: Formal power series

4.3 Degree, Leading Coefficients, and Integral Domains

Definition 4.3.1 Degree, Leading Coefficient, Monic
Theorem 4.3.1 Degree of a Sum
Theorem 4.3.2 Degree of a Product
Definition 4.3.2 Integral Domain
Example Which Rings Are Integral Domains
Corollary 4.3.3 Polynomial Rings over Integral Domains
Proposition 4.3.4 Units of a Polynomial Ring
Definition 4.3.3 Polynomials of Degree at Most n
Example Membership in Polynomials of Bounded Degree
Definition 4.3.4 Polynomials of Degree Exactly n
Example Comparing the Two Sets
Exercise A1
Exercise B1: Put the proof in order
Exercise B2: Units in polynomial rings
Exercise B3: Counting polynomials over a finite field
Exercise C1: Exit check: polynomial rings over integral domains
Exercise C2: Find the error

4.4 Evaluation, Substitution, and Polynomial Functions

Definition 4.4.1 Evaluation, Root, Polynomial Function
Theorem 4.4.1 Evaluation Respects the Operations
Definition 4.4.2 Ring Homomorphism
Example Ring Homomorphisms and Two Impostors
Theorem 4.4.2 Substitution Principle
Corollary 4.4.3 Changing the Coefficient Ring
Definition 4.4.3 Composition of Polynomials
Proposition 4.4.4 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

4.5 The Formal Derivative

Definition 4.5.1 Formal Derivative
Example Formal derivatives in five rings
Theorem 4.5.1 Rules for the Formal Derivative
Proposition 4.5.2 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

5.1 Division with Remainder

Definition 5.1.1 Divisibility of Polynomials
Proposition 5.1.1 Rules for Divisibility of Polynomials
Theorem 5.1.2 Division with Remainder in $F[x]$
Example Long division in two fields
Theorem 5.1.3 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

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

Definition 5.2.1 Ideal Generated by Polynomials
Theorem 5.2.1 Ideals of $F[x]$ Are Principal
Definition 5.2.2 Greatest Common Divisor of Polynomials
Theorem 5.2.2 Bézout’s Theorem for Polynomials
Definition 5.2.3 Coprime Polynomials
Corollary 5.2.3 Coprime Means 1 Is a Combination
Lemma 5.2.4 One Step of the Euclidean Algorithm
Theorem 5.2.5 The Euclidean Algorithm for Polynomials
Example A run of the Euclidean algorithm
Proposition 5.2.6 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

5.3 Irreducible Polynomials and Unique Factorization

Definition 5.3.1 Irreducible Polynomial
Proposition 5.3.1 Irreducibility in Low Degree
Theorem 5.3.2 Euclid’s Lemma for Polynomials
Corollary 5.3.3 An Irreducible Dividing a Product
Proposition 5.3.4 Coprime Divisors of a Polynomial
Lemma 5.3.5 An Irreducible Factor Exists
Theorem 5.3.6 Unique Factorization of Polynomials
Example The Factorizations of $x^4 - 1$
Theorem 5.3.7 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

5.4 Roots, Multiplicity, and the Formal Derivative

Theorem 5.4.1 Counting Roots
Theorem 5.4.2 Polynomials Determined by Values
Example Polynomial Functions over a Finite Field
Theorem 5.4.3 Expansion in Powers of $x - c$
Definition 5.4.1 Multiplicity of a Root
Lemma 5.4.4 Multiplicity through the Cofactor
Proposition 5.4.5 Multiplicity of a Product
Theorem 5.4.6 Splitting Off All Roots
Definition 5.4.2 Splitting over a Field
Example Multiplicities by Repeated Division
Theorem 5.4.7 Repeated Roots and the Derivative
Corollary 5.4.8 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

5.5 Interpolation, Quotient Rings, and the Chinese Remainder Theorem

Definition 5.5.1 Lagrange Polynomials
Theorem 5.5.1 Lagrange Interpolation
Example A cubic through four points
Definition 5.5.2 Congruence Modulo a Polynomial
Theorem 5.5.2 The Ring of Polynomials Modulo m
Theorem 5.5.3 The Quotient Is a Field Exactly When m Is Irreducible
Example A field with four elements, and the complex numbers again
Lemma 5.5.4 Coprime to Each Factor, Coprime to the Product
Theorem 5.5.5 Chinese Remainder Theorem for Polynomials
Proposition 5.5.6 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

5.6 The Fundamental Theorem of Algebra

Lemma 5.6.1 Growth of a Complex Polynomial
Lemma 5.6.2 A Lipschitz Bound for a Complex Polynomial
Theorem 5.6.3 The Modulus of a Complex Polynomial Attains Its Minimum
Lemma 5.6.4 A Direction with Negative Real Part
Lemma 5.6.5 d’Alembert’s Lemma
Example One step of descent
Theorem 5.6.6 Fundamental Theorem of Algebra
Corollary 5.6.7 Complex Polynomials Split
Example Factoring a quartic over the reals and over the complex numbers
Proposition 5.6.8 Conjugate Roots of Real Polynomials
Corollary 5.6.9 The Irreducible Real Polynomials
Theorem 5.6.10 Factorization of Real Polynomials
Corollary 5.6.11 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