AG01 What is Abstract Algebra?

Abstract Algebra is a study of algebraic structures such as groups, rings, and fields. Groups are mathematician's approach to symmetry, while rings provide a way to do number theory with sets of numbers larger than the integers. The Abstract Algebra project and the common approaches to algebraic structures is discussed. The historical tidbits include origins of the word "Algebra", the solving of cubic, quartic, and quintic equations, connection of rings & Diophantine equations, Euler's conjectures in line with Fermat's last theorem, and classical straightedge and compass constructions (doubling the cube, trisecting an angle, squaring a circle). Subscribe @Shahriari for math videos for undergraduates.
00:00 Introduction
00:13 Abstract Algebra as a coherent subject & Plan for this video
02:24 Groups, Rings, and Fields as Algebraic Structures
03:25 The Abstract Algebra project
04:05 Why study Abstract Algebraic Structures?
05:39 Objections to the project
06:16 To prove only one group with 167 elements...
06:46 Common Approaches in Abstract Algebra
08:00 Each algebraic structure is different
10:55 Groups & Symmetry
11:53 History: the quadratic equation
12:54 History: Origins of "Algebra"
14:10 History: Solving Cubic and Quartic equations
16:20 History: Groups & The Quintic
18:07 Group Theory & A Problem on Bijections
19:34 Rings
24:02 History: Euler's Conjectures
25:34 Fields
26:16 History: Straightedge and Compass constructions
27:25 Classical Problems: Can you double a cube, trisect an angle, square a circle?
28:08 Field theory and high school algebra
28:47 The Plan going forward
This is the introductory video in a series of videos for a full course on undergraduate abstract algebra based on my book:

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA, USA.

[PatrickL]

Whoa, this guy seems like he knows a lot about algebra