Math 5286H: Fundamental Structures of Algebra II (Spring 2012)

Lectures: MWF 10:10-11:00 in Vincent Hall 213.

Instructor: Gregg Musiker (musiker "at" math.umn.edu)

Office Hours: M 1:25-2:25, WF 11:00-12:05 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 Sylow Theorems

Spring) Rings, fields, modules, and Galois theory

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: Algera, by Michael Artin, (2nd edition 2011, Prentice Hall).

Fall: some (but not all) of Chapters 1-7;

Spring: some (but not all) of Chapters 11-16

Note that I will be following the brand new second edition in this course, and encourage you to do the same. If you have the first edition, your text will more or less contain the same topics (although with occasional exceptions and sometimes in a different order), but the references to section numbers and exercises will not match the ones I announce in class and on the website. If you are in this situation, please find a classmate with the new edition or come talk to me early on.

Other useful texts (On reserve in math library)
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:

  • Homework (50%): There will be 5 homework assignments due approximately every other week (tentatively) on Fridays in class. The first homework assignment is due on 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 24th) and March 30th (due April 6th). 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 on April 27th, to be turned in during class on May 4th.

  • 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.

    Course Syllabus and Tentative Lecture Schedule

  • (Jan 18) Lecture 1: Definition of a Ring and Polynomial Rings (Artin, Sec. 11.1-11.2)
  • (Jan 20) Lecture 2: Ring Homorphism (Artin, Sec. 11.3)
  • (Jan 23) Lecture 3: Ideals (Artin, Sec. 11.3)
  • (Jan 25) Lecture 4: More on Ideals and Quotient Rings (Artin, Sec. 11.4)
  • (Jan 27) Lecture 5: Correspondence Theorems and Adjoining Elements (Artin, Sec. 2.10, 11.4-11.5)
  • (Jan 30) Lecture 6: More on Adjoining Elements and Product Rings (Artin, Sec. 11.6)
  • (Feb 1) Lecture 7: Further examples of Ideals in Z[x] (Artin, Sec. 11.5-11.6)
  • (Feb 3) Lecture 8: Fractions (Artin, Sec. 11.7)
  • (Feb 6) Lecture 9: Maximal Ideals (Artin, Sec. 11.8)
  • (Feb 8) Lecture 10: Factoring Integers and Unique Factorization Domains (Artin, Sec. 12.1-12.2)
  • (Feb 10) Lecture 11: More on Euclidean Domains, UFD's, and PID's (Artin, Sec. 12.2)
  • (Feb 13) Lecture 12: Gauss's Lemma (Artin, Sec. 12.3)
  • (Feb 15) Lecture 13: Factoring Integer Polynomials (Artin, Sec. 12.4)
  • (Feb 17) Lecture 14: Algebraic Integers and Gauss Primes (Artin, Sec. 12.5, 13.1-13.2)
  • (Feb 20) Lecture 15: Ideals in Z[\sqrt{-5}] and Ideal Multiplication (Artin, Sec. 13.3-13.4)
  • (Feb 22) Lecture 16: Factoring Ideals and Prime Ideals (Artin, Sec. 13.5-13.6)
  • (Feb 24) Lecture 17: Ideal classes and Class Groups (Artin, Sec. 13.7-13.8)
  • (Feb 27) Lecture 18: Field Extensions (Artin, Sec. 15.1-15.2)
  • (Feb 29) Lecture 19: The Degree of a Field Extension (Artin, Sec. 15.3)
  • (Mar 2) Lecture 20: Finding the Irreducible Polynomial (Artin, Sec. 15.4)
  • (Mar 5) Lecture 21: Ruler and Compass Constructions I (Artin, Sec. 15.5)
  • (Mar 7) Lecture 22: Ruler and Compass Constructions II and Adjoining Rotts (Artin, Sec. 15.5-15.6)
  • (Mar 9) Lecture 23: Separability and Proof of the Primitive Element Theorem (Artin, Sec. 15.6, 15.8)
  • (Mar 12 - 16): Spring Break
  • (Mar 19) Lecture 24: Finite Fields (Artin, Sec. 15.7)
  • (Mar 21) Lecture 25: Algebraic Closures and the Fundamental Theorem of Algebra (Artin, Sec. 15.10)
  • (Mar 23) Lecture 26: Introduction to Galois Theory and Symmetric Functions (Artin, Sec. 16.1)
  • (Mar 26) Lecture 27: More on Symmetric Functions and Discriminants (Artin, Sec. 16.1-16.2)
  • (Mar 28) Lecture 28: More on Splitting Fields and Isomorphisms of Fields Extensions (Artin, Sec. 16.3-16.4)
  • (Mar 30) Lecture 29: Fixed Fields and Galois Extensions (Artin, Sec. 16.5-16.6)
  • (April 2) Lecture 30: The Main Theorem of Galois Theory (Artin, Sec. 16.6-16.7)
  • (April 4) Lecture 31: Cubic Extensions and History of Cardano's Formula (Artin, Sec. 16.8)
  • (April 6) Lecture 32: Quartic Extensions (Artin, Sec. 16.9)
  • (April 9) Lecture 33: Quartic Extensions II (Artin, Sec. 16.9)
  • (April 11) Lecture 34: Roots of Unity (Artin, Sec. 16.10)
  • (April 13) Lecture 35: Cyclotomic Extensions II (Artin, Sec. 16.10)
  • (April 16) Lecture 36: Kummer Extensions (Artin, Sec. 16.11)
  • (April 18) Lecture 37: Quintic Extensions (Artin, Sec. 16.12)
  • (April 20) Lecture 38: Modules and Free Modules (Artin, Sec. 14.1-14.2)
  • (April 23) Lecture 39: Diagonalizing Integer Matrices (Artin, Sec. 14.3-14.4)
  • (April 25) Lecture 40: Generators and Relations, Noetherian Rings (Artin, Sec. 14.5-14.6)
  • (April 27) Lecture 41: Structure Theorem for Rings over PID's and Critical Groups of Graphs (Artin, Sec. 14.7)
  • (April 30) Lecture 42: More on Critical Groups of Graphs (Related Slides)
    David Perkinson's Webpage on Sandpiles
  • (May 2) Lecture 43: Jordan Canonical Form II (Artin, Sec. 14.8)
  • (May 4) Lecture 44: A Glimpse of Representation Theory (Artin, Chapter 10)


  • Homework assignments and exams
    Assignment or Exam Due date Problems from Artin text,
    unless otherwise specified
    Homework 1 Friday 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 Friday 2/17 Chapter 11: 6.2, 6.3, 6.4, 6.5, 6.7, 6.8, 7.1, 7.2, 7.3, 8.2, 8.3,
    Chapter 12: 1.4, 2.1, 2.3, 2.6, 3.1, 3.2
    Exam 1 Friday 2/24 Download Here
    Homework 3 Friday 3/9 Mostly computational this homework; more theoretical problems next set.
    Chapter 13: 1.1, 3.2 (a-b), 4.1, 5.1, 5.2 (Main Lemma is Lemma 13.4.8), 6.1, 6.3, 7.1, 7.4,
    Chapter 15: 2.1, 2.2, 3.2, 3.3, 3.7, 4.2 (a, b, d)
    Homework 4 Friday 3/30 Wednesday 4/4 Chapter 15: 5.1, 6.3, 7.4, 7.5, 7.7, 7.8, 8.1
    Chapter 16: 1.3, 2.2, 3.1, 3.2 (b-c), 4.1, 6.2, 6.3
    Exam 2 Friday 4/6 Friday 4/13 Download Here
    Homework 5 Friday 4/27 Chapter 16: 9.3, 9.6, 9.8, 9.11, 10.3, 12.1, 12.3, 12.5, 12.7, (Bonus): M.12
    Chapter 14: 1.3, 1.4, 2.2
    Final Exam Friday 5/4 Download Here