Math 627B: Modern Algebra II
Spring 2006

Instr: Mike O'Sullivan



Assignments

Due dates may change depending on schedule.

Assignment Due date Read Turn in.
I Wed. 1/25 IVA I.1 #1-5
IVA I.2 #1-9, 13-15
IVA I.1 #5
IVA I.2 #2, 4d, 5, 6, 8, 13, 15d
II Mon. 2/6 IVA I.3 #1-6
IVA I.4 #1-12
IVA I.5 #1, 3-9
IVA I.3 #4, 5, 6
IVA I.4 #5, 8, 9, 11 (or 12)
IVA I.5 #14, 15, 16
III Fri. 2/17 IVA II.1 #1-3
IVA II.2 #1-7, 10-12
IVA II.3 #1-10
IVA II.1 #1d, 2b, 3b, 3c
IVA II.2 #1b, 12 Probs. A, B, C, D
IVA II.3 #5, 6, 9
IV Wed. 3/1 IVA I.6 #1-12
IVA II.7 #1-11
IVA II.8 #1-7
IVA II.9 #1-8, 12, 13
IVA II.6 #10, 12
IVA II.7 #2b, 3b
IVA II.8 #6, 7
IVA II.9 #2, 3, 12
V Mon. 3/27 IVA III.1 #1-9
IVA III.2 #5
IVA III.3 #1-15
IVA III.5 #2,3, 6-16
IVA III.6 #1-5, 8-13


IVA III.3 (one of #6, 7, 8, 9), (one of #14, 15, or Bezier cubic)
IVA III.5 #13, 16
IVA III.6 #9, 11
VI Mon. 4/17 IVA IV.1 #5-9
IVA IV.2 #8-10, 13, 15
IVA IV.3 #6-15
IVA IV.4 #1-4, 8, 9, 10
IVA IV.5 #1-5, 12, 6-16
IVA IV.6 #1-4
IVA IV.2 #15
IVA IV.3 #11, 12
IVA IV.4 #3, 4
IVA IV.6 #7
Extra problems:

A. Give the grevlex and the grlex orderings for n=3 and for monomials up to degree 2. This is enough to show they are different orderings.

B. Let n=3 . For each of the monomial orderings lex, grlex and grevlex orders, find vectors u_1, u_2, u_3 which determine that ordering.

C. Find a total ordering on N_0^3 which is determined by a single vector u_1 .

D. What condition(s) on the u_i are necessary to ensure that a group ordering on N_0^n is also a well-ordering and therefore a monomial ordering?

E. (optional) Prove one of the steps in Robbiano's classification of group orderings discussed in class on Mon. 2/6.