Abstract Algebra | Axiom Academy

1

Introduction to Algebraic Structures

Master introduction to algebraic structures

16 modules

1

Symmetries of FieldsINTRO

2

The Ultimate Algebraic StructureINTRO

3

The Group AxiomsLESSON

4

Group or Not Group?LESSON

5

Matrix Groups GL₂(ℝ) and SL₂(ℝ)LESSON

6

Order of ElementsLESSON

7

Foundation SummaryLESSON

8

Ring or Not Ring?LESSON

9

Verifying 2Z is a RingEXAMPLE

10

Fields SummaryLESSON

11

Extension Field BuilderLESSON

12

Algebraic Closure DefinitionLESSON

13

Verifying ⟨(123)⟩ is a Subgroup of S₄EXAMPLE

14

Verifying ⟨x² + 1⟩ is an Ideal in R[x]EXAMPLE

15

Verifying A₄ is Normal in S₄EXAMPLE

16

Fields ReferenceFORMULA SHEET

2

Groups

Master groups

139 modules

1

The Language of MathematicsINTRO

2

The Power of OneINTRO

3

The Group ZooINTRO

4

Groups Within GroupsINTRO

5

Partitioning GroupsINTRO

6

The Great CorrespondenceINTRO

7

When Galois Groups are LargestINTRO

8

Building Blocks of Abelian GroupsINTRO

9

Prime Power SubgroupsINTRO

10

Groups in MotionINTRO

11

The Art of RearrangementINTRO

12

When Cosets Form GroupsINTRO

13

Structure-Preserving MapsINTRO

14

Group FingerprintingINTRO

15

Sets, Functions, and RelationsLESSON

16

Foundation SkillsPRACTICE

17

Modular Arithmetic WorldLESSON

18

Group Detective LabLESSON

19

Elementary TheoremsLESSON

20

Elementary Properties PracticePRACTICE

21

Group GalleryLESSON

22

Proving (ab)⁻¹ = b⁻¹a⁻¹EXAMPLE

23

Group VerificationPRACTICE

24

Symmetry HunterLESSON

25

The Dihedral Group D₄LESSON

26

Computing in Z₄ × Z₆EXAMPLE

27

Computing Cosets of ⟨r²⟩ in D₄EXAMPLE

28

Order CalculatorLESSON

29

Finding a Primitive Element of GF(8)EXAMPLE

30

Generator LaboratoryLESSON

31

Cyclic Groups ZₙLESSON

32

Cyclic Group StructureLESSON

33

Verifying Group Structure for (Z₆, +)EXAMPLE

34

Cyclic Group ComputationsPRACTICE

35

Subgroups of Cyclic GroupsLESSON

36

The Subgroup TestLESSON

37

Subgroup ExplorerLESSON

38

Finding Generators of Z₂₀EXAMPLE

39

Computing Gal(Q(√2, √3)/Q)EXAMPLE

40

Subgroup IdentificationPRACTICE

41

Center and CentralizersLESSON

42

Subgroups SummaryLESSON

43

Left and Right CosetsLESSON

44

Coset CalculatorLESSON

45

Lagrange's TheoremLESSON

46

Cayley's Theorem via Regular ActionEXAMPLE

47

Applications of LagrangeLESSON

48

FTGT ApplicationsPRACTICE

49

Lagrange ApplicationsPRACTICE

50

Groups of Prime OrderLESSON

51

Groups of Order 6LESSON

52

Small Group IdentifierLESSON

53

Identifying GroupsPRACTICE

54

Small Group ClassificationPRACTICE

55

Homomorphism BuilderLESSON

56

Properties of HomomorphismsLESSON

57

Kernel and Image DefinitionsLESSON

58

Finding Kernel and Image of detEXAMPLE

59

Normal SubgroupsLESSON

60

Quotient Group ConstructionLESSON

61

Computing S₃/A₃EXAMPLE

62

Quotient CalculatorLESSON

63

Isomorphism DetectiveLESSON

64

Computing DiscriminantsEXAMPLE

65

Homomorphism VerificationPRACTICE

66

Recognizing IsomorphismsLESSON

67

Second and Third Isomorphism TheoremsLESSON

68

Applying FTGT to Q(√2, √3)/QEXAMPLE

69

Isomorphism Theorems PracticePRACTICE

70

Homomorphism SummaryLESSON

71

Verifying φEXAMPLE

72

Symmetric and Alternating GroupsLESSON

73

The Symmetric Group S₃EXAMPLE

74

Conjugacy Classes in SₙLESSON

75

The 15-Puzzle GroupLESSON

76

Group Actions DefinitionLESSON

77

Group Actions PracticePRACTICE

78

Action ExplorerLESSON

79

The Sylow TheoremsLESSON

80

Sylow Theory ApplicationsPRACTICE

81

Sylow CalculatorLESSON

82

Finding Sylow Subgroups of S₄EXAMPLE

83

Groups of Order pqLESSON

84

Classifying Groups of Order 15EXAMPLE

85

Classifying Groups of Order 4EXAMPLE

86

External Direct ProductsLESSON

87

Direct Products and Abelian GroupsPRACTICE

88

Direct Product LaboratoryLESSON

89

Fundamental Theorem of Finite Abelian GroupsLESSON

90

Classifying Abelian Groups of Order 36EXAMPLE

91

Abelian Group ClassifierLESSON

92

Advanced Groups SummaryLESSON

93

Ring AxiomsLESSON

94

Ring CalculatorLESSON

95

Zero Divisors and UnitsLESSON

96

Ideal BuilderLESSON

97

Ring Homomorphism LaboratoryLESSON

98

Ring Theory SummaryLESSON

99

Polynomial Ring MasteryPRACTICE

100

Field ExplorerLESSON

101

Finite Field LaboratoryLESSON

102

Proving Z₅ is a FieldEXAMPLE

103

Multiplicative Group of Finite FieldsLESSON

104

Galois Fields PreviewLESSON

105

Extension Field ChallengeLESSON

106

Field Automorphism DefinitionLESSON

107

The Galois GroupLESSON

108

Finding Automorphisms of Q(∛2)EXAMPLE

109

Automorphism PracticePRACTICE

110

Galois Group CalculatorLESSON

111

Fixed FieldsLESSON

112

Fundamental Theorem PreviewLESSON

113

Galois Extension CheckerLESSON

114

Fundamental Theorem of Galois TheoryLESSON

115

Correspondence LaboratoryLESSON

116

Finding All Subfields of Q(ω₈)EXAMPLE

117

Normal Subgroups and Galois ExtensionsLESSON

118

Galois Extensions PracticePRACTICE

119

Normal Subgroups and QuotientsPRACTICE

120

Solvable GroupsLESSON

121

Proving x⁵ - 4x + 2 is Not SolvableEXAMPLE

122

Radical ExtensionsLESSON

123

Galois Theorem on SolvabilityLESSON

124

Solvability DetectiveLESSON

125

Solvability PracticePRACTICE

126

Cyclotomic ExtensionsLESSON

127

Kummer ExtensionsLESSON

128

Inverse Galois ProblemLESSON

129

Galois Theory SummaryLESSON

130

Verifying Q(ω)/Q is GaloisEXAMPLE

131

The Unity of Abstract AlgebraLESSON

132

Groups Foundation ReferenceFORMULA SHEET

133

Galois Theory ReferenceFORMULA SHEET

134

Advanced Groups ReferenceFORMULA SHEET

135

Subgroups ReferenceFORMULA SHEET

136

Groups Foundation AssessmentQUIZ

137

Advanced Group Theory AssessmentQUIZ

138

Homomorphisms and Quotients AssessmentQUIZ

139

Subgroups and Cyclic Groups AssessmentQUIZ

3

Subgroups and Cosets

Master subgroups and cosets

9 modules

1

Galois Extension DefinitionLESSON

2

Normal ExtensionsLESSON

3

Quotient Ring BuilderLESSON

4

Quotient Ring ConstructionLESSON

5

Computing Z[x]/⟨x² + 1⟩EXAMPLE

6

Orbits and StabilizersLESSON

7

Constructing GF(4)EXAMPLE

8

Applying First Isomorphism to φEXAMPLE

9

Homomorphism ReferenceFORMULA SHEET

4

Homomorphisms and Isomorphisms

Master homomorphisms and isomorphisms

19 modules

1

Structure-Preserving Ring MapsINTRO

2

The Fundamental Homomorphism TheoremINTRO

3

Homomorphism Definition and ExamplesLESSON

4

The Homomorphism Obstacle CourseLESSON

5

Injectivity and SurjectivityLESSON

6

Kernel-Image ExplorerLESSON

7

First Isomorphism TheoremLESSON

8

Isomorphism Theorem LaboratoryLESSON

9

Automorphism ExplorerLESSON

10

Ideals - Special SubringsLESSON

11

Ring Homomorphism DefinitionLESSON

12

The Evaluation HomomorphismEXAMPLE

13

Ring Homomorphism PropertiesPRACTICE

14

Kernel and Image for RingsLESSON

15

Kernel and Image ComputationsPRACTICE

16

Chinese Remainder TheoremLESSON

17

Classification of Finite FieldsLESSON

18

Ring Theory ReferenceFORMULA SHEET

19

Ring Theory AssessmentQUIZ

5

Quotient Groups

Master quotient groups

6 modules

1

Collapsing Ideals to ZeroINTRO

2

Maximal and Prime IdealsLESSON

3

Quotient Field PropertiesLESSON

4

Quotient Ring ComputationsPRACTICE

5

Domain DetectiveLESSON

6

Division Algorithm in F[x]EXAMPLE

6

Permutation Groups

Master permutation groups

4 modules

1

Cycle CalculatorLESSON

2

Permutation ComputationsPRACTICE

3

Equivalence Relations and PartitionsEXAMPLE

4

Decomposing (1 3 5 2)(2 4 3) into Disjoint CyclesEXAMPLE

7

Rings

Master rings

23 modules

8

Ideals and Quotient Rings

Master ideals and quotient rings

2 modules

1

Principal Ideal DomainsLESSON

2

Finding All Ideals of Z₁₂EXAMPLE

9

Fields

Master fields

15 modules

1

Algebraically Complete FieldsINTRO

2

Fields with Finitely Many ElementsINTRO

3

Finite Field ApplicationsLESSON

4

Finite Field ApplicationsLESSON

5

Finite Field MasteryPRACTICE

6

Subfields of Finite FieldsLESSON

7

Classical ConstructionsLESSON

8

Simple ExtensionsLESSON

9

Algebraic Extensions PracticePRACTICE

10

Field Extensions F ⊆ KLESSON

11

The Field of Algebraic NumbersEXAMPLE

12

Building Q from ZEXAMPLE

13

Field Extension ComputationsPRACTICE

14

Irreducible PolynomialsLESSON

15

Constructing Q(√2)EXAMPLE

10

Polynomial Rings

Master polynomial rings

2 modules

1

Solving Polynomial EquationsINTRO

2

Separable ExtensionsLESSON

11

Galois Theory and Applications

Master galois theory and applications

3 modules

1

Galois Theory in ActionINTRO

2

Advanced Galois PracticePRACTICE

3

Galois Theory AssessmentQUIZ