Math 5286H: Fundamental Structures of Algebra II (Spring 2020)
Lectures: MWF 10:10-11:00 in Vincent Hall 2.
Instructor: Gregg Musiker (musiker "at" math.umn.edu)
Office Hours: Mondays 3:30-4:30, Wednesdays 11:00-12:00, Fridays 2:30-3:30 in Vincent 251; also by appointment.
Fall: Vector spaces, linear algebra, group theory, symmetry, and the Spectral Theorem
Spring: Simple Groups, the Sylow Theorems, Rings, fields, and Galois theory. Modules if time permitting
Prerequisites:
Some previous exposure to linear algebra (vectors, matrices, determinants) (such as from 2243, 2373, or 2573) would help.
Also, one should either have the ability to write and read mathematical proofs (such as from 2283, 2574, or 3283), or have the desire and drive to learn how.
Required text:
Algbera, by Michael Artin, (2nd edition 2011, Prentice Hall).
Fall: some (but not all) of Chapters 1-8;
Spring: some (but not all) of Chapters 7, 11-16
Homework (50%):
There will be 5 homework assignments due approximately every other week in class. The due dates are listed below, and the first homework assignment is due on Monday February 3rd.
I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained; the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.
Exams (15% each):
There will be 2 take-home exams, handed out on February 17th (due February 24st) and March 27th (due April 3rd). Each will be open book, open notes, and with calculators allowed. However, for these exams, you are not allowed to consult other electronic sources, such as the internet, and you are not allowed to collaborate or consult with other students or other human sources. These exams are to be collected in class.
Final Exam (20%):
The final exam will be take-home as well, under the same policies as above, handed out by April 27th, to be turned in during class on May 4th.
(Jan 22) Lecture 1: Welcome Back and Definition of a Ring and Polynomial Rings (Artin 11.1-11.2)
(Jan 24) Lecture 2: More on Polynomial Rings and Ring Homomorphisms (Artin 11.2-11.3)
(Jan 27) Lecture 3: The Substitution Principle and Ideals (Artin 11.3)
(Jan 29) Lecture 4: Quotient Rings (Artin 11.4)
(Jan 31) Lecture 5: Adjoining Elements (Artin 11.5)
(Feb 3) Lecture 6: Integral Domains and Product Rings (Artin 11.6-11.7)
(Feb 5) Lecture 7: Idempotents and Fractions (Artin 11.6-11.7)
(Feb 7) Lecture 8: Maximal Ideals and Algebraic Geometry (Artin 11.8-11.9)
(Feb 10) Lecture 9: Factoring Integers and Unique Factorization Domains (Artin 12.1-12.2)
(Feb 12) Lecture 10: Euclidean Domains and Gaussian Integers (Artin 12.2)
(Feb 14) Lecture 11: Gauss Primes and Factoring Polynomials (Artin 12.2, 12.5)
(Feb 17) Lecture 12: Gauss' Lemma and Factoring Integer Polynomials (Artin 12.3-12.4)
(Feb 19) Lecture 13: Quadratic Number Fields: Algebraic Integers and their Factorizations (Artin 13.1-13.2)
(Feb 21) Lecture 14: More on Rings of Integers in Quadratic Imaginary Number Fields (Artin 13.1-13.2)
(Feb 24) Lecture 15: Multiplication of Ideals in Z[sqrt(-5)] (Artin 13.3-13.4)
(Feb 26) Lecture 16: Factoring Ideals and Prime Ideals (Artin 13.5-13.6)
(Feb 28) Lecture 17: Ideal Classes and Glimpse of Class Groups (Artin 13.7-13.8)
(Mar 2) Lecture 18: Examples of Fields and Algebraic and Transcendental Elements (Artin 15.1-15.2)
(Mar 4) Lecture 19: Degrees of Field Extensions (Artin 15.3)
(Mar 6) Lecture 20: Corollaries to the Multiplicative Property of Degree and Finding the Irreducible Polynomial (Artin 15.3-15.4)
(Mar 9) Spring Break
(Mar 11) Spring Break
(Mar 13) Spring Break
(Mar 16) In-service Teaching Day
(Mar 18) Discusion of Resuming Course and Lecture 21: Ruler and Straightedge Constructions (Artin 15.5)
(Mar 20) Lecture 22: Ruler and Straightedge Constructions II (Artin 15.5)
(Mar 23) Lecture 23: Ruler and Straightedge Constructions III and Adjoining Roots (Artin 15.6)
(Mar 25) Lecture 24: Finite Fields I (Artin 15.7)
(Mar 27) Lecture 25: Finite Fields II, Primitive Elements and The Fundamental Theorem of Algebra (Artin 15.7, 15.8, 15.10)
(Mar 30) Lecture 26: Review of Group Actions and Class Equation (Artin 7.1-7.4)
(Apr 1) Lecture 27: Simple Groups, Alternating Groups, and Statement of the Sylow Theorems (Artin 7.5-7.7)
(Apr 3) Lecture 28: Applications of the Sylow Theorems and Groups of Order 12 (Artin 7.7-7.8)
(Apr 6) Lecture 29: Proofs of the Sylow Theorems (Artin 7.7)
(Apr 8) Lecture 30: Symmetric Functions (Artin 16.1)
(Apr 10) Lecture 31: Discriminants (Artin 16.2)
(Apr 13) Lecture 32: More on Splitting Fields and Isomorphisms of Field Extensions (Artin 16.3-16.4)
(Apr 15) Lecture 33: Fixed Fields and Galois Extensions (Artin 16.5-16.6)
(Apr 17) Lecture 34: The Main Theorem of Galois Theory (Artin 16.7)
(Apr 20) Lecture 35: Proof of the Main Theorem of Galois Theory and Applications (Artin 16.7)
(Apr 22) Lecture 36: Cubics and Quartics via Galois Theory (Artin 16.8-16.9)
(Apr 24) Lecture 37: Cubics and Quartics via Galois Theory II (Artin 16.8-16.9)
(Apr 27) Lecture 38: Cyclotomic Extensions and Kummer Extensions (Artin 16.10 -16.11)
(Apr 29) Lecture 39: Qunitic Polynomials (Artin 16.12)
(May 1) Lecture 40: Glimpse of Modules and the Structure Theorem for Abelian Groups and PIDs (Artin Chapter 14)
(May 4) Lecture 41: TBA
S/N Grade: If you are registered S/N, I will submit a grade of S if your letter grade is C or above, and otherwise a grade of N.
Incomplete grade: A grade of 'I' will only be considered when failure to complete all course requirements is for reasons beyond the student's control.
The minimum requirement for an incomplete grade is a substantial amount of course work completed at the level of C or better. An 'I' grade also requires a
written agreement between the student and the instructor on how the missing work will be completed. After one year, an 'I' turns into an 'F' if the course work
is not completed. Any arrangement for an incomplete grade can only be considered before the final exam.
Disability Accommodations: Disability services promotes access and equity for everyone at the U of M. If you are registered with DS and would like to
discuss accommodations, please contant the instructor as soon as possible. If you require accommodations, but are not registered with DS, please contact their
office.
To add or drop the course: For the various rules and deadlines for adding or dropping the course, visit One-Stop.
Scholastic Misconduct: Academic dishonesty in any work for this course will be grounds for awarding a grade of 'F' for the entire course. University
policies regarding academic dishonesty, credit, workload expectations, and grading standards are available
here and here.
Instructor: Gregg Musiker (musiker "at" math.umn.edu)
Office Hours: Mondays 3:30-4:30, Wednesdays 11:00-12:00, Fridays 2:30-3:30 in Vincent 251; also by appointment.
Course Description:
This is the second semester of a course in the basic algebra of groups, rings, fields, and vector spaces. Roughly speaking, the Fall and Spring semesters should divide the topics as follows:Fall: Vector spaces, linear algebra, group theory, symmetry, and the Spectral Theorem
Spring: Simple Groups, the Sylow Theorems, Rings, fields, and Galois theory. Modules if time permitting
Prerequisites:
Fall: some (but not all) of Chapters 1-8;
Spring: some (but not all) of Chapters 7, 11-16
| Title | Author(s) | Year |
|---|---|---|
| Introduction to abstract algebra | W. Keith Nicholson | 2007 |
| Contemporary abstract algebra | Joseph A. Gallian | 1994 |
| Algebra | Thomas Hungerford | 1980 |
| Abstract Algebra | Dummit and Foote | 2004 |
| Abstract algebra: theory and applications | Thomas Judson | 1994 |
Grading:
I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained; the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.
Class Participation:
Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead.Tentative Lecture Schedule
| Assignment or Exam | Due date | Problems from Artin text, unless otherwise specified |
|---|---|---|
| Homework 1 | Monday 2/3 | Chapter 11: # 1.1, 1.3, 1.6 (a), 1.8, 1.9, 3.2, 3.3, 3.8, 3.9, 3.12, 3.13, 4.1, 4.2, 4.3, 5.1, 5.4 |
| Homework 2 | Monday 2/17 | Chapter 11: # 6.4, 6.8, 7.1, 7.2, 7.3, 7.5, 8.2, 8.3, 8.4 Chapter 12: # 2.1, 2.6, 2.7, 2.8, 4.7, 5.1, 5.3 |
| Exam 1 | Monday 2/24 | |
| Homework 3 | Friday 3/6 | Chapter 11: # 1.2, 8.1 Chapter 12: # 3.1, 3.2, 4.1, 4.12 Chapter 13: # 3.2, 4.1, 5.1, 5.2, 7.1, 7.4 |
| Homework 4 | |
Chapter 15: # 2.2, 2.3, 3.1, 3.2, 3.3, 3.4, 3.7, 3.8, 5.1, 6.3, 7.4, 7.5, 7.6, 7.10, 7.14, 8.1 |
| Exam 2 | |
|
| Homework 5 | |
Chapter 7: # 4.7, 7.3, 7.4, 7.9, 7.10, 8.5, 8.6 Chapter 16: # 2.2, 4.1, 6.2, 6.3, 7.1, 7.4, 7.11, 9.11 |
| Final Exam | Monday 5/4 | |