# Math 5286: Honors fundamental structures of algebra

## Spring 2008

 Prerequisites: Math 5285 or its equivalent: exposure to some abstract group theory and linear algebra, along with the ability to write and read mathematical proofs. Instructor: Victor Reiner (You can call me "Vic"). Office: Vincent Hall 256 Telephone (with voice mail): 625-6682 E-mail: reiner@math.umn.edu Classes: Mon-Wed-Fri 10:10-11:00am, Vincent Hall 207. Office hours: Monday 12:20pm, Tuesday 3:35pm, Friday 11:15am; also by appointment. Course content: This is the second semester of a course in the basic algebra of groups, ring, fields, vector spaces, and perhaps modules over rings. Roughly speaking, the Fall semester (Math 5285) covered vector spaces, linear algebra, group theory and symmetry, doing a lot of Chapters 1-6 of the course text by Artin, touching lightly upon Chapters 7 and 9. In this second semester, we will study more seriously rings, fields, and perhaps modules over rings, covering some portion of Chapters 10-14 of Artin's text; see below for what portion more specifically. To give some feeling for the topics, rings and fields are the keys to understanding why it is that... you can't trisect an angle using only a straightedge and compass, unlike the formulae known for solving quadratic, cubic and 4th degree polynomial equations (see here for example), there are no such formulae in general for 5th and higher degree equations, while understanding modules helps show that even though complex matrices can't always be diagonalized, they can get to something pretty close called Jordan canonical form, finite abelian groups are always direct products of cyclic groups. Required text: Algebra, by Michael Artin, Prentice-Hall, 1991. We expect to cover much of Chapters 10-14, in this order: Ch. 10 (skip Sec. 7,8) Ch. 11 (skip Sec. 5-12) Ch. 13 (skip Sec. 7,8,9) Ch. 14 (skip Sec. 6) Ch. 12 (skip Sec. 8)
Other useful texts
Level Title Author(s), Publ. info Location
Lower A concrete introduction
to higher algebra
Childs, Springer-Verlag 1995 On reserve in math library
Lower Contemporary abstract algebra Gallian, Houghton-Mifflin 1998 On reserve in math library
Same Topics in algebra Herstein, Wiley & Sons 1999 On reserve in math library
Higher Abstract algebra Dummit and Foote, Wiley & Sons 2004 On reserve in math library
 Homework, exams, grading: As with the Fall semester, there will be 5 homework assignments due usually every other week, but 2 weeks where there will be a week-long take-home midterm exam, a week at the end with a week-long take-home final exam. Tentative dates for the assignments and exams are in the schedule below. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. The take-home midterms and final exam are open-book, open-library, open-web, but in contrast to the homework on exams, no collaboration or consultation of human sources is allowed. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom near Vincent Hall 105. 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. Final course grade basis : Homework = 50% of grade Each of 2 midterms = 15% of grade Final exam = 20% of grade
Homework assignments
Assignment or Exam Due date Problems from Artin,
unless otherwise specified
Homework 1 2/13 10.1: 4,5,6,8,12
10.2: 7
10.3: 7,8,9,15,17,24
10.4: 3(b) (removed 2)
10.5: 1,12 (removed 3,10; moved 5 to HW2)
Homework 2 2/27 10.5: 5,14,15
10.6: 2,3,5
Chap. 10 Misc. Probs.: 2
11.1: 1,4,8,15 (removed 12)
11.2: 5,8,13
11.3: 4
11.4: 3,4,8
Exam 1 3/5 Midterm exam 1 in PostScript, PDF.
Homework 3 3/26 13.2: 3(a,b),4,5
13.3: 1,3(a,b,c),8,14
13.4: 1,5,6
13.6: 10 (moved 3,5,8,9,11 to HW4)
Chap. 13 Misc. Probs.: 2
Homework 4 4/9 13.5: 2
13.6: 3,5,8,9,11
14.1: 7, 8(a), 9, 12, 13
14.5: 2
Exam 2 4/16 Midterm exam 2 in PostScript, PDF.
Homework 5 4/30 14.1: 1,15,17 (problem 20 removed from HW)
14.5: 11 (problem 4 removed from HW)
14.9: 8
12.1: 1, 6, 7
12.2: 3,4,5
Final Exam (Friday!) 5/9 Final exam in PostScript, PDF.
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