5530.001,  Spring 2000
Algebra II
Conley

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INSTRUCTOR:   Charles  Conley,  GAB  475,  (940)  565 - 3326

OFFICE HOURS:   Monday and Wednesday,  11:30-12:30 and 2:00-3:00

CLASS MEETS:   Monday and Wednesday, 12:30 - 1:50, GAB 406

FINAL EXAM DATE AND TIME:   Monday,  May 8,  10:30 - 12:30.  MIDTERM:   Wednesday, March 8

TEXT:   For Rings: Chapter 2 of Basic Algebra, by Jacobson, 2nd edition.  For Modules: Part III of Abstract Algebra, by Dummit and Foote.  For Galois Theory: Abstract Algebra, by Fraleigh.

This course will cover rings, modules, and fields.  We will begin with rings: definitions, examples, homomorphisms, ideals, quotient rings, commutative domains, fraction fields, polynomial rings and their quotients, PIDs, and UFDs.  During this unit we will emphasize the examples of Z, Z[x], Q[x], and the finite fields.

Our main goals in the modules section will be the fact that subgroups of lattices are lattices, and the rational canonical form for matrices over Q.  We will also discuss the computation of the Jordan canonical form.

We will use the remaining time to begin discussing the Galois theory of fields.  This section will be continued in an informal seminar during August, aimed at preparing for the qualifying exam.

It is the responsibility of students with certified disabilities to provide the instructor with appropriate documentation from the Dean of Students Office.

FINAL  EXAM:  Monday,  May 8, 10:30 - 12:30.  It will be drawn almost entirely from old qualifying exam problems.
You will be given a list of about 30 problems that may be on the test in advance.  The test will be two sets of 4 each:
you will have to do 2 from each set.  It will not be comprehensive; it will only cover material from after the midterm.

HOMEWORK  12,  Due Wednesday, May 3 (may be turned in late) (from Dummit and Foote):

12.3:   1, 2, 5, 9-11, 18, 22-24

HOMEWORK  11,  Due Wednesday, April 26 (may be turned in late) (from Dummit and Foote):

12.2:   1, 3, 4, 6-11, 14

HOMEWORK  10,  Due Wednesday, April 19 (from Dummit and Foote):

12.1:   1, 2, 4, 5, 13, 14

Also, find a "good" basis of  Z^2,  with respect to the submodules
generated by each of  the following sets of  vectors:

{3e_1,  6e_2},  {4e_1,  6e_2},  {5e_1 + 5e_2},  {2e_1 + 2e_2,  5e_1 + 5e_2},

{e_1 + e_2,  e_1 - e_2},  and  {2e_1 + e_2,  -e_1 + 2e_2}.

HOMEWORK  9,  Due Wednesday, April 12 (from Dummit and Foote):

10.3:   6-7, 9-11, 15;   11.1:   1-2, 6-9.

Notes:  You may do 11.1.9 for k=2 and 3 only if you like.  11.1.6 is very tricky.

HOMEWORK  8,  Due Wednesday, April 5 (from Dummit and Foote):

10.1:   1, 3-9, 11a;   10.2:   1, 2, 4-6, 9.

HOMEWORK  7,  Due Wednesday, March 29:

2.16:   1-6.

MIDTERM  EXAM:  Wednesday,  March 8.  It will be drawn almost entirely from old qualifying exam problems.
You will be given a list of about 40 problems that may be on the test in advance.  The test will be two sets of 4 each:
you will have to do 2 from each set.

HOMEWORK  6,  Due Monday, March 6:

2.14:   1, 4, 6, and the first two sentences of 2,

2.15:   1, 4, 12, 2*

HOMEWORK  5,  Due Wednesday, February 23:

2.10:  4 (for r = 2 and 3 only);     2.11:   1-6,  9-10,  14,

Recommended, but do not turn in:   0.6:   1, 3;     2.11:   12, 13.

HOMEWORK  4,  Due Wednesday, February 16:

2.9:   1, 2, 4, 5*;      2.10:  1, 2, 3 (for r = 1 only).

HOMEWORK  3,  Due Wednesday, February 9:  (NOTE:  Section 2.9 is postponed until next week)

2.7:  1, 2, 4, 5, 10*, 11 (see problem 9 for definitions), 12*, 15, 16, 17,

Note: 2.7: 4, 10, 12, 15, 16 and 2.9.5 are all important theorems, which you
will need to know later on.  I will give hints for them.

HOMEWORK  2,  Due Wednesday, February 2:

2.5:  2, 3, 4, 5, 6, 7;      2.6:  1, 2, 3.

HOMEWORK  1,  Due Wednesday, January 26:

2.1:  2, 4;      2.2:  2, 3, 4**, 6*;      2.3:  1, 3, 4, 6, 7, 8

(in 2.3: 3, 4, 6, 7, 8, let the ring R be the complex numbers).