**5530.001, Spring 2000**

**Algebra II**

**Conley**

the University of North Texas. |

**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.

I

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).