Field Theory: We consider the property of algebraic in terms of finite degree, and we define algebraic numbers as those complex numbers that are algebraic over the rationals. Then we give an overview of algebraic numbers with examples.

1. Mystery Division Problem
2. GT1. Definition of Group
3. GT1.1. Example of Group Inverse
4. Order 2 Elements in Finite Group
5. Example of Group Cancellation Law
6. GT2. Definition of Subgroup
7. Example of Group: GL(2, R) (1 of 3)
8. GT3. Cosets and Lagrange's Theorem
9. GT4. Normal Subgroups and Quotient Groups
10. GT5. Index 2 Theorem and Dihedral Groups
11. Example of Group: GL(2,R) (2 of 3)
12. Example of Group: GL(2, R) (3 of 3)
13. GT6. Centralizers, Normalizers, and Direct Products
14. GT7. The Commutator Subgroup
15. GT8. Group Homomorphisms
16. GT9. Group Isomorphisms
17. Example of Group Isomorphism
18. GT10. Examples of Non-Isomorphic Groups
19. GT11. Group Automorphisms
20. GT11.1. Automorphisms of A4
21. GT12. Aut(Z/n) and Fermat's Little Theorem
22. GT12.1. Automorphisms of Dihedral Groups
23. GT13. Groups of Order 8
24. GT14. Semidirect Products
25. GT15. Group Actions
26. GT16. Cayley's Theorem
27. GT16.1 Examples of Cayley's Theorem
28. GT17. Symmetric and Alternating Groups
29. Order 12 Subgroups in S5
30. GT17.1. Permutation Matrices
31. |Z(G)| for |G|=pq
32. GT18. Conjugacy and The Class Equation
33. Class Equation for Dihedral Group D8
34. GT18.1. Class Equation for Dihedral Groups
35. GT18.2. A_n is Simple (n ge 5)
36. GT19. Cauchy's Theorem
37. GT20. Overview of Sylow Theory
38. GT20.1. Sylow Theorems - Proofs
39. GT20.2 Sylow Theory for Simple 60
40. Sylow Theory for Order 12 Groups 1
41. Sylow Theory for Order 12 Groups 2
42. Simple Group 168 - Sylow Theory - Part 1
43. Simple Group 168 - Sylow Theory - Part 2
44. GT21. Internal Products
45. GT22. The Fundamental Theorem of Finite Abelian Groups
46. GT23. Composition and Classification
47. RNT1.1. Definition of Ring
48. RNT1.2. Definition of Integral Domain
49. RNT1.2.1. Example of Integral Domain
50. RNT1.2.2. Order of a Finite Field
51. RNT1.3. Ring Homomorphisms
52. RNT1.4. Ideals and Quotient Rings
53. RNT1.4.1. Example of Quotient Ring
54. RNT2.1. Maximal Ideals and Fields
55. RNT2.1.1. Finite Fields of Orders 4 and 8
56. RNT2.2. Principal Ideal Domains
57. RNT2.3. Euclidean Domains
58. RNT2.3.1. Euclidean Algorithm for Gaussian Integers
59. RNT2.4. Gaussian Primes
60. RNT2.5. Polynomial Rings over Fields
61. RNT2.5.1. Euclidean Algorithm for Z/3[x]
62. RNT2.6.1. Gauss' Lemma
63. RNT2.6.2. Eisenstein's Criterion
64. FIT1.1. Number Fields
65. FIT1.2. Characteristic p
66. FIT2.1. Field Extensions
67. FIT2.2. Simple Extensions
68. FIT2.2.1. Example: Cubic Extension
69. FIT2.2.2. Example: Quartic Extension
70. FIT2.3.1. Algebraic Numbers
71. FIT2.3.2. Cardinality and Transcendentals
72. FIT2.3.3. Algebraic Extensions
73. FIT3.1.1. Roots of Polynomials
74. FIT3.1.2. Roots of Real Polynomials
75. FIT3.1.3. Example of Splitting Field
76. FIT3.1.4. Factoring Example: Artin-Schreier Polynomials
77. FIT3.2.1. Cyclotomic Polynomials
78. FIT3.2.2. Mobius Inversion Formula
79. FIT4.1. Galois Group of a Polynomial
80. FIT4.2. Automorphisms and Degree
81. FIT4.3. Galois Correspondence 1 - Examples
82. FIT4.3.1. Galois Group of Order 8
83. FIT4.3.2. Example of Galois Group over Finite Field

Includes course on Group Theory (problems and solutions at website) and Ring Theory, and Field Theory. For Prerequisites on proofs and sets, see the Math Major Basics course.