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Austin Christian

Georgia Tech Mathematics

GT Summer Edge Program: Algebra (2023)

Instructor: Austin Christian

office hours: TWF 1pm-2pm, Skiles 022

TA: Joshua Marsh

office hours: R 12pm-1pm, Skiles 201

Meeting time: TWRF 9:00am-9:50am

Meeting location: Skiles 114

a gif showing the transformations r, ar, rar, and arar applied to a triangle

Announcements

For at least the first few weeks of the summer, our primary text will be Judson's Abstract Algebra: Theory and Applications. You are encouraged to familiarize yourself with chapters 1 and 2 before we get started, and should complete the six exercises listed below before our first day.

Schedule

The following is a tentative, nonbinding schedule. It will be updated as the summer progresses. You should attempt all listed exercises, and submit solutions to those with stars.

	Topics covered	Notes	Exercises
Week 1 (5/30-6/2)	T: Introductions, summer goals, groups W: Groups: basic properties, important examples, subgroups R: Subgroup criteria, cyclic subgroups F: Cyclic subgroups, permutation groups	May 30 May 31 June 1 June 2	Due 5/30: The following exercises from Judson. 1.4: 25, 29 2.4: 1, 5, 16, 17, 20 Due 6/2: The following exercises from Judson. 3.5: 27, 28, 31, 48, 53 4.5: 1, 6, 10, 13, 20, 31, 46
Week 2 (6/6-6/9)	T: Subgroups of \(S_n\), cosets W: Theorems of Lagrange, Euler, and Fermat R: Homomorphisms, isomorphisms, and Cayley's theorem F: External and internal direct products	June 6 June 7 June 8	Due 6/6: The following exercises from Judson. 5.4: 1, 3, 4, 5, 12, 14, 23, 31, 32, 33, 34 Due 6/9: The following exercises from Judson. 5.4: 35* 6.5: 3, 4, 6, 9, 12, 15, 16, 21, 23 9.4: 1, 2, 3, 4, 8
Week 3 (6/13-6/16)	T: Normal subgroups W: Simplicity of the alternating group R: The first isomorphism theorem F: More isomorphism theorems	June 13 June 14 June 15 June 16	Due 6/13: The following exercises from Judson. 9.4: 9, 10, 11, 14-17, 18, 19, 37, 41, 42 Due 6/16: The following exercises from Judson. 10.4: 2, 3, 4, 8, 9, 10, 13, 14 11.4: 2, 5, 6, 7, 9, 10, 14, 16, 19
Week 4 (6/20-6/23)	T: Classification of finite abelian groups W: Composition series R: The Jordan-Hölder theorem F: Group actions	June 20 June 21 June 22 June 23	Due 6/20: The following exercises from Judson. 11.5: 2, 3, 4, 5, 6, 7, 9-11, 12 12.4: 15(a)* Due 6/25: The following exercises from Judson. 13.4: 3, 6, 7, 9, 17, 19, 20 14.5: 1-4, 5, 7
Week 5 (6/27-6/30)	T: Burnside's counting theorem W: The Sylow theorems R: The Sylow theorems, day 2 F: Midterm exam	June 27 June 28 June 29	Due 6/29: The following exercises from Judson. 14.5: 8, 9, 20, 21, 23
Week 6 (7/4-7/7)	T: Fourth of July W: Rings, integral domains, fields R: Subrings and ideals F: Maximal and prime ideals	July 5 July 6 July 7	Due 7/4: The following exercises from Judson. 15.4: 5, 6, 7, 8, 13, 16, 17, 19, 21, 22 Due 7/7: The following exercises from Judson. 16.7: 2, 5, 8, 12, 16, 20, 26, 27, 28, 33, 37
Week 7 (7/11-7/14)	T: Polynomial rings, day 1 W: Polynomial rings, day 2 R: Fields, day 1 F: Fields, day 2	July 11 July 12 July 13 July 14	Due 7/11: The following exercises from Judson. 16.7: 4, 5, 7, 8, 9, 39 17.5: 3, 4, 5, 7, 10 Due 7/14: The following exercises from Judson. 17.5: 11, 14, 18, 20, 21, 25, 26, 28 21.5: 1, 3, 4, 5, 11, 12, 14
Week 8 (7/18-7/21)	T: Galois theory, day 1 W: Galois theory, day 2 R: Galois theory, day 3 F: Final exam	July 18 July 19 July 20	Due 7/18: The following exercises from Judson. 21.5: 9, 15, 22, 23 Due 7/20: The following exercises from Judson. 23.5: 1, 2, 5, 6, 7, 11, 14, 20