MTH 461-D: Survey of Modern Algebra

University of Miami, Spring 2021

Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: 10-11 Monday and Wednesday, or by appointment
Each office hour session is on Zoom now.
Time and Location: 11:45-12:35 MWF in Stubblefield 508.

Course syllabus can be found here.

Final Project Description: here

Final Exam Problems: here

Course Schedule

Date	Lecture Content	Reading
1/25/21	"Lecture 0": Syllabus overview. Introduction to course material.
1/27/21	Definition of a Group. First examples of groups. Abelian groups. Cayley tables.	Lecture 1 notes
1/29/21	Basic properties of groups. Subgroups.	Lecture 2 notes
2/1/21	Integers mod n. Review of equivalence classes. Cyclic groups.	Lecture 3 notes
2/3/21	More modular arithmetic. The group Z_n^x.	Lecture 4 notes
2/5/21	Orders of group elements.	Lecture 5 notes
2/8/21	Symmetries and permutations.	Lecture 6 notes
2/10/21	Quiz for half of class. Review of quiz.	Quiz 1 solutions
2/12/21	More on symmetric groups. Even and odd permutations.	Lecture 7 notes
2/15/21	Alternating groups. Symmetries of the tetrahedron.	Lecture 8 notes
2/17/21	Cosets and Lagrange's Theorem.	Lecture 9 notes
2/19/21	Consequences of Lagrange's Theorem. Euler's Theorem. Fermat's Little Theorem.	Lecture 10 notes
2/22/21	Applications to cryptography. RSA cryptosystem.	Lecture 11 notes
2/24/21	Quiz for half of class. Review of quiz.	Quiz 2 solutions
2/26/21	Normal subgroups and quotient groups.	Lecture 12 notes
3/1/21	Homomorphisms and isomorphisms.	Lecture 13 notes
3/3/21	Wellness Wednesday. Recorded lecture on complex numbers in group theory.	Lecture 14 notes
3/5/21	More isomorphisms, and kernels.	Lecture 15 notes
3/8/21	First Isomorphism Theorem, and symmetries of a cube.	Lecture 16 notes
3/10/21	More Isomorphism Theorems.	Lecture 17 notes
3/12/21	Cayley's Theorem.	Lecture 18 notes
3/15/21	Classification of finite groups.	Lecture 19 notes
3/17/21	Quiz for half of class. Review of quiz.	Quiz 3 solutions
3/19/21	Rubik's cube group.	Lecture 20 notes
3/22/21	Defintion of a Ring.	Lecture 21 notes
3/25/21	More ring basics. Homomorphisms. Quaternions.	Lecture 22 notes
3/27/21	Kernels, ideals, and quotient rings.	Lecture 23 notes
3/29/21	Principal ideals.	Lecture 24 notes
3/31/21	Principal ideal domains.	Lecture 25 notes
4/2/21	Rings and geometry. Varieties.	Lecture 26 notes
4/5/21	Prime and maximal ideals.	Lecture 27 notes
4/7/21	Quiz.
4/9/21	Euclidean domains and UFDs.	Lecture 28 notes
4/12/21	Primes as the sum of two squares.	Lecture 29 notes
4/14/21	Wellness Wednesday.
4/16/21	Constructions of Fields.	Lecture 30 notes
4/19/21	Algebraic field extensions.	Lecture 31 notes
4/21/21	Degrees of field extensions.	Lecture 32 notes
4/23/21	Straightedge and compass constructions.	Lecture 33 notes
4/26/21	Splitting fields and Galois groups.	Lecture 34 notes
4/28/21	Quiz.
4/30/21	The Galois Theorem.	Lecture 35 notes

Homework Assignments

Assignment	Due Date	Remarks
Homework 1: PDF file	February 5	solutions
Homework 2: PDF file	February 12	solutions
Homework 3: PDF file	February 19	solutions
Homework 4: PDF file	February 26	solutions
Homework 5: PDF file	March 5	solutions
Homework 6: PDF file	March 12	solutions
Homework 7: PDF file	March 19	solutions
Homework 8: PDF file	March 22	solutions
Homework 9: PDF file	April 9
Homework 10: PDF file	April 30