# Abstract Algebra: Groups, Rings & Fields

### Course Description

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.

Math Doctor Bob in Lecture 66: Field Extensions
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### Video Lectures & Study Materials

# Lecture Play Lecture
1 Mystery Division Problem (8:58) Play Video
I. Group Theory
2 GT1. Definition of Group (15:00) Play Video
3 GT1.1. Example of Group Inverse (5:33) Play Video
4 Order 2 Elements in Finite Group (7:39) Play Video
5 Example of Group Cancellation Law (6:28) Play Video
6 GT2. Definition of Subgroup (17:12) Play Video
7 Example of Group: GL(2, R) (1 of 3) (7:10) Play Video
8 GT3. Cosets and Lagrange's Theorem (16:34) Play Video
9 GT4. Normal Subgroups and Quotient Groups (20:37) Play Video
10 GT5. Index 2 Theorem and Dihedral Groups (16:58) Play Video
11 Example of Group: GL(2,R) (2 of 3) (8:25) Play Video
12 Example of Group: GL(2, R) (3 of 3) (12:22) Play Video
13 GT6. Centralizers, Normalizers, and Direct Products (14:05) Play Video
14 GT7. The Commutator Subgroup (15:21) Play Video
15 GT8. Group Homomorphisms (17:41) Play Video
16 GT9. Group Isomorphisms (24:00) Play Video
17 Example of Group Isomorphism (8:33) Play Video
18 GT10. Examples of Non-Isomorphic Groups (19:51) Play Video
19 GT11. Group Automorphisms (25:02) Play Video
20 GT11.1. Automorphisms of A4 (11:01) Play Video
21 GT12. Aut(Z/n) and Fermat's Little Theorem (17:38) Play Video
22 GT12.1. Automorphisms of Dihedral Groups (25:58) Play Video
23 GT13. Groups of Order 8 (18:02) Play Video
24 GT14. Semidirect Products (18:42) Play Video
25 GT15. Group Actions (20:19) Play Video
26 GT16. Cayley's Theorem (22:04) Play Video
27 GT16.1 Examples of Cayley's Theorem (9:50) Play Video
28 GT17. Symmetric and Alternating Groups (20:53) Play Video
29 Order 12 Subgroups in S5 (10:38) Play Video
30 GT17.1. Permutation Matrices (21:10) Play Video
31 |Z(G)| for |G|=pq (6:42) Play Video
32 GT18. Conjugacy and The Class Equation (22:05) Play Video
33 Class Equation for Dihedral Group D8 (15:57) Play Video
34 GT18.1. Class Equation for Dihedral Groups (10:49) Play Video
35 GT18.2. A_n is Simple (n ge 5) (16:30) Play Video
36 GT19. Cauchy's Theorem (15:00) Play Video
37 GT20. Overview of Sylow Theory (14:37) Play Video
38 GT20.1. Sylow Theorems - Proofs (15:38) Play Video
39 GT20.2 Sylow Theory for Simple 60 (19:58) Play Video
40 Sylow Theory for Order 12 Groups 1 (11:31) Play Video
41 Sylow Theory for Order 12 Groups 2 (9:07) Play Video
42 Simple Group 168 - Sylow Theory - Part 1 (11:13) Play Video
43 Simple Group 168 - Sylow Theory - Part 2 (8:24) Play Video
44 GT21. Internal Products (20:10) Play Video
45 GT22. The Fundamental Theorem of Finite Abelian Groups (8:57) Play Video
46 GT23. Composition and Classification (17:25) Play Video
II. Ring Theory
47 RNT1.1. Definition of Ring (16:18) Play Video
48 RNT1.2. Definition of Integral Domain (12:55) Play Video
49 RNT1.2.1. Example of Integral Domain (7:07) Play Video
50 RNT1.2.2. Order of a Finite Field (5:47) Play Video
51 RNT1.3. Ring Homomorphisms (14:03) Play Video
52 RNT1.4. Ideals and Quotient Rings (20:47) Play Video
53 RNT1.4.1. Example of Quotient Ring (5:07) Play Video
54 RNT2.1. Maximal Ideals and Fields (18:55) Play Video
55 RNT2.1.1. Finite Fields of Orders 4 and 8 (10:37) Play Video
56 RNT2.2. Principal Ideal Domains (15:22) Play Video
57 RNT2.3. Euclidean Domains (20:26) Play Video
58 RNT2.3.1. Euclidean Algorithm for Gaussian Integers (6:39) Play Video
59 RNT2.4. Gaussian Primes (22:31) Play Video
60 RNT2.5. Polynomial Rings over Fields (18:49) Play Video
61 RNT2.5.1. Euclidean Algorithm for Z/3[x] (9:06) Play Video
62 RNT2.6.1. Gauss' Lemma (16:11) Play Video
63 RNT2.6.2. Eisenstein's Criterion (7:25) Play Video
III. Field Theory
64 FIT1.1. Number Fields (14:24) Play Video
65 FIT1.2. Characteristic p (11:26) Play Video
66 FIT2.1. Field Extensions (12:49) Play Video
67 FIT2.2. Simple Extensions (12:40) Play Video
68 FIT2.2.1. Example: Cubic Extension (7:47) Play Video
69 FIT2.2.2. Example: Quartic Extension (8:11) Play Video
70 FIT2.3.1. Algebraic Numbers (13:22) Play Video
71 FIT2.3.2. Cardinality and Transcendentals (8:56) Play Video
72 FIT2.3.3. Algebraic Extensions (16:13) Play Video
73 FIT3.1.1. Roots of Polynomials (21:12) Play Video
74 FIT3.1.2. Roots of Real Polynomials (17:14) Play Video
75 FIT3.1.3. Example of Splitting Field (7:20) Play Video
76 FIT3.1.4. Factoring Example: Artin-Schreier Polynomials (7:21) Play Video
77 FIT3.2.1. Cyclotomic Polynomials (19:43) Play Video
78 FIT3.2.2. Mobius Inversion Formula (16:45) Play Video
79 FIT4.1. Galois Group of a Polynomial (22:51) Play Video
80 FIT4.2. Automorphisms and Degree (14:47) Play Video
81 FIT4.3. Galois Correspondence 1 - Examples (19:36) Play Video
82 FIT4.3.1. Galois Group of Order 8 (10:50) Play Video
83 FIT4.3.2. Example of Galois Group over Finite Field (6:10) Play Video

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