**3510.001, Fall 2003**

**Introduction to Abstract Algebra**

**Conley**

the University of North Texas. |

**INSTRUCTOR:** Charles Conley, GAB 419, (940) 565-3326

**OFFICE HOURS:** MW 12:30-2:00, F 12:30-1:30

**CLASS MEETS:** MWF 11:00-11:50, GAB 206

**EXAMS, HOMEWORK, AND GRADING:** There will be two 100 point
midterms, on Wednesday, Sept. 24 and Wednesday, Oct. 29, and a
comprehensive 180 point final, on Friday, Dec. 12, 10:30-12:30.
There will be thirteen homeworks, due at the beginning of the last day
of class each week. They will be worth 10 points each, excepting
those due the first week and the week of Thanksgiving, which will be
worth 5 points each. There will be no make-up exams except for
emergencies, and
late homework will be worth half-credit.

**TEXT AND PREREQUISITES: **The text is *A first course in
Abstract Algebra,* 6th edition, by J. Fraleigh. The
prerequisite is Math 2510-20, the analysis sequence.

**TOPICS:** We will cover two main topics, groups and
rings. The section of the course on groups will be an
introduction to their structure, with an emphasis on examples such as
the cyclic and dihedral groups, the groups of symmetries of the
Platonic solids, and the symmetric groups. The section on rings
will begin with arithmetic in the integers, the unique factorization
theorem, and modular arithmetic. Then we will discuss more
general rings, emphasizing the particular
case of polynomial rings and their quotients, which lead to field
theory. Prior experience with analytic proofs will be assumed;
here you will
be introduced to algebraic proofs.

FINAL EXAM: Friday, Dec.12, 10:30-12:30

HOMEWORK 13, due Friday, December 5

Section 5.3: 1-10, 24-28

Extra Problems:

(A) Find an isomorphism from the additive group Z_{16}to
the multiplicative group Z_{17}^*.

(B) Find all generators of Z_{17}^*

(C) Find 1/6 in Z_{17}

(D) Find all square roots of 2 in Z_{17}

(E) Find all square roots of 6 in Z_{17}

HOMEWORK 12, due Wednesday, November 26 (This homework is
worth only
5 points, not the usual 10)

Section 5.2: 2-4, 8-10, 12, 27

HOMEWORK 11, due Friday, November 21

Section 5.1: 2, 4, 6, 12, 14-16, 19, 28, 32, 38, 40, 49, 52, 55

HOMEWORK 10, due Friday, November 14

Section 3.1: 25, 28, 29, 33-38, 41, 42, 49

Section 3.2: 31, 34*

EXTRA PROBLEMS:

In problems (A)-(D), prove that no surjective
homomorphism exists from

(A) S_3 to Z_3, (B)
A_4 to Z_2, (C) A_4 to any 4 element group, (D) S_4 to any 8
element group.

In problems (E)-(H), argue geometrically that surjective
homomorphisms exist and compute their kernels:

(E) S_3 to Z_2, (F) A_4 to Z_3, (G) S_4 to S_3, (H) S_4
to S_2.

HOMEWORK 9, due Friday, November 7

Section 2.4: 1-5, 9, 11, 15-20

Section 3.1: 4, 5, 16-21, 44, 45, 47

Extra problems: These are postponed
until next week.

EXAM 2: Wednesday, October 29

HOMEWORK 8, due Friday, October 24

Section 0.2: 34-36

Section 2.3: 1-4, 6-12, 16, 26-30, 33, 35, 38, 39

HOMEWORK 7, due Friday, October 17

Section 2.1: 18-20, 23-26, 42-44

Section 2.2: 1-3, 7-18, 24ab, 31

HOMEWORK 6, due Friday, October 10

Section 1.5: 16-20, 34-39, 52-54, 64, 65a

Section 2.1: 1-9, 11-13, 16-17

HOMEWORK 5, due Friday, October 3

Section 1.4: 36, 41, 42, 46, 47, 49, 55

Section 1.5: 7, 9, 11, 12-15, 21, 22, 26-33

EXAM 1: Wednesday, September 24

HOMEWORK 4, due Friday, September 19

Section 1.3: 20a, 26, 27, 29, 30, 32, 33, 34, 35, 40

Section 1.4: 5, 7, 20, 27, 28

HOMEWORK 3, due Friday, September 12

Section 1.2: 16, 17, 25, 26, 29, 30, 31

Section 1.3: 1-6, 10

HOMEWORK 2, due Friday, September 5

Section 1.1: 1-7

Section 1.2: 2-7, 10

HOMEWORK 1, due Friday, August 29:

Section 0.1: 29, 30

Section 0.2: 3

Section 0.3: 1, 3, 4, 5, 8b