Math 547/702I: Algebraic Structures II.

Spring 2010

University of South Carolina

      Instructor  
      Matthew Boylan  
  Office     LeConte 400G  
  Phone     777-8874  
  E-mail     boylan@math.sc.edu  
  Office hours     Mon. 1:30 - 2:30  
  Tues. 2 - 3  
  Fri. 9 - 10  
  And by appointment  

  • University of South Carolina Mathematics Department .

    • Course information :

  • Prerequisite : Math 546.

  • Text : Abstract Algebra, by Beachy and Blair, 3rd ed., Waveland Press (2006).

  • Course objectives : Students will master concepts and solve problems on rings: ideals, polynomial rings, Euclidean domains, unique factorization domains; fields: extensions, Galois theory, Euclidean constructions: modules over principal ideal domains.

  • Class schedule :

      Lecture     MWF     10:10 - 11:00     LeConte 303B  

  • Homework : Problem sets will be assigned each week and will be due the following week. They may be turned in during class or to my mailbox any time before 3 pm on the due date. Some of the assigned problems will be graded. You will be graded on both the correctness of your solutions and on the quality of your exposition (i.e., how well you write the solution). You are encouraged to discuss problems with other students in the class. However, you must write solutions to the problems yourself (in your own words).

  • Exam schedule : (3 one-hour exams and a cumulative final exam)

      Exam 1:     Wednesday     February 10     10:10 - 11:00     LeConte 303B  
      Exam 2:     Friday     March 19     10:10 - 11:00     LeConte 303B  
      Exam 3:     Friday     April 16     10:10 - 11:00     LeConte 303B  
      Final Exam:     Tuesday     April 28     9:00 - 12:00     LeConte 303B  

    Missed exams will not be made up. Exceptions may be made for documented illness/family emergency.

  • Calculators (and computer software such as Maple) may be used on homework unless otherwise noted. They may not be used on exams. Moreover, they will be needed neither for homework nor exams.

  • Grading :

          Points/650     % of grade  
      Three hour exams:     100 pts. x 3     15% x 3  
      Final exam:     200 pts.     30%  
      Homework:     150 pts.     25%  
      Total:     650 pts.     100%  

    Graduate credit: Students enrolled in 702I are expected to do a term project, due at the end of the semester.

    Extra credit will not be offered.

    Letter grades will be assigned according to the following approximate scale unless otherwise noted:

      A     B     C     D     F  
      90 - 100     80 - 90     70 - 80     60 - 70     0 - 60  

    Note: The deadline to drop without a WF is Monday, February 22.

  • Other policies : Cell phones and other electronic devices are to be turned off or put on silent or vibrate mode during class meetings. Homework and exam grades may be discussed only up to 7 days after being returned.

  • Other help resources : Math Lab (free drop-in tutoring at various locations on campus, including LeConte 105), Private tutors , On-line materials for courses   (547 websites from past semesters), Math library .

  • Syllabus (which contains all of the above information): (pdf)

    • Lectures and Homework :

    Lectures Homework
      Dates     Sections     Topics     Problems     HW due date  
      January  
      Week 1     11     M     5.1     Commutative rings.     5.1: 1 b, d, 2 a, b, e, f, 4, 7, 8, 14, 15; extra: 11. Homework 1. Solutions.     1/22.  
      13     W         Integral domains, fields, quaternions.          
      15     F     5.2     Ring homomorphisms.     5.2: 2, 3, 5, 10, 11, 13, 19, 22; extra: 14, 15, 23.Homework 2. Solutions.     2/1.  
      Week 2     20     W     5.3     Ideals.          
      22     F         Factor rings.          
      Week 3     25     M         Principal ideals.          
      27     W         Prime and maximal ideals.          
      29     F     9.1     Principal ideal domains.     5.3: 7, 8, 9, 11, 13, 14, 17 b, c, 20; extra: 12, 21, 26 (all parts but (a)). Homework 3. Solutions.     2/10.  
      February  
      Week 4     1     M                  
      3     W         Euclidean domains.          
      5     F         Associates, irreducibles, primes, gcds.          
      Week 5     8     M                  
      10     W     Exam I.     Covers: 5.1, 5.2, 5.3, 9.1; homework 1, 2, 3; class notes: 1/11 - 2/3.     Summary notes; Solutions.      
      12     F         Unique factorization domains.     9.1: 3, 10, 13; 9.2: 6; extra: 9.1: 11, 14. Homework 4. Solutions.     2/22.  
      Week 6     15     M         Ascending chain condition; maximal condition.          
      17     W         PID => UFD.          
      19     F         Chinese Remainder Theorem.          
      Week 7     22     M         Polynomial rings.          
      24     W         Polynomial rings over fields are Euclidean.          
      26     F         Gauss' Lemma.     4.2: 4 (any 2 of a, b, c, d), 5 (any 2 of a, b, c, d), 7, 15, 16; 5.3: 24; extra: 4.2: 13, 14. Homework 5.Solutions.     3/15.  
      March  
      Week 8     1     M         Z[x] is a UFD.          
      3     W         Eisenstein's Criterion.          
      5     F         Remainder, Factor, and Rational Root Theorems.          
      Week 9     15     M         Polynomial rings in several variables, power series rings, local rings.          
      17     W         Derivatives and multiplicities of roots of polynomials.          
      19     F     Exam II.     Covers: Homework 4, 5; class notes: 2/3 - 3/5.     Summary notes. Solutions.      
      Week 10     22     M         Review of vector spaces.          
      24     W     6.1     Algebraic and transcendental numbers.     1 c, e, f, 2, 3, 4, 5, 9, 10, 11. Homework 6. Solutions.     4/2.  
      26     F     6.1     The minimal polynomial.          
      Week 11     29     M     6.2     The degree of a number field.          
      31     W     6.2              
      April  
      2     F     6.2              
      Week 12     5     M     6.4     The splitting field of a polynomial.          
      7     W     6.4         6.1: 12; 6.2: 1 a, b, c, d, 3, 4, 5, 7, 9, 10. Homework 7. Solutions.     4/16.  
      9     F     8.1     Introduction to Galois theory.          
      Week 13     12     M     8.1              
      14     W     8.2     Multiplicities of roots.          
      16     F     Exam III.     Covers: Homework 6, 7; class notes: 3/15 - 4/12.     Summary notes. Solutions.      
      Week 14     19     M                  
      21     W         Normal extensions.     6.4: 1, 2, 4, 7, 14; 8.1: 4, 6; 8.2: 5; 8.3: 3, 4, 5. Homework 8. Solutions.      
      23     F         The Galois correspondence.          
      Week 15     26     M         The Fundamental Theorem of Algebra.          
      Finals week     28     W     Guide.    Final Exam, 9 - 12.     Cumulative. Solutions.