MATH
421 – Abstract Algebra
Spring 2011
Professor:
Dr. Janet Vassilev
Office: SMLC 324
Office
Hours: MWF 2-3 pm and by appointment.
Telephone: 
(505)
277-2214
email: jvassil@math.unm.edu
webpage: http://www.math.unm.edu/~jvassil
Text
:
A First Course in Abstract Algebra, 7th Edition By
John Fraleigh.
Course
Meetings: The course lectures will be held in SMLC 356 on Mondays, Wednesdays
and Fridays at 1-1:50 pm.
Topics: Theory of fields, algebraic field extensions, advanced group theory
and Galois theory.
Daily
Quizzes (100 points): The first 5 minutes of everyday (excluding
exam days) will consist of an open notebook quiz on the concepts of the
previous lecture. The quizzes will be
worth 5 points each. I will drop your 4
lowest quizzes and average the remaining quizzes to obtain a score out of 100.
Homework
(200 points): Homework will be assigned weekly on Wednesdays and will
be collected the following Wednesday at the beginning of class. Homework
will not be graded unless it is written in order and labeled
appropriately. The definitions and theorems in the text and
given in class are your tools for the homework proofs. If the theorem has a name, use it. Otherwise, I would prefer that you fully
describe the theorem in words that you plan to use, than state “by Theorem
3”. Each week around 4 of the assigned problems will be graded. The
weekly assignments will each be worth 20 points. I will drop your lowest two homework scores
and the remaining homework will be averaged to get a score out of 200.
Exams (400
points): I will give two midterms (100 points) and a final (200
points). There are no make up exams. If a test is missed,
notify me as soon as possible on the day of the exam.
For the midterms only, if you have a legitimate and documented
excuse, your grade will be recalculated without that test. The Midterms
are tentatively scheduled for Monday, February 21 and Friday, April 1.
The Final is on Friday, May 13, from 12:30 pm-2:30 pm.
Grades: General guidelines
for letter grades (subject to change; but they won't get any more strict):
90-100% - A; 80-89% - B; 70-79% - C; 60-69% - D; below 60% - F. In
assigning Final Grades for the course, I will compare your grade on all course
work (including the Final) and your grade on the Final Exam. You will
receive the better of the two grades.
Tentative
Schedule (for Dr. Vassilev’s Abstract Algebra):
Date
|
Section
|
Topic
|
Homework
|
1/19
|
23
|
Review of
factoring polynomials
|
|
1/21
|
23
|
Factoring
polynomials
|
Quiz 2: What is an irreducible polynomial in F[x]?
|
1/24
|
27
|
Ideals in F[x]
|
Quiz 3: What is the content of a polynomial f(x) in
Q[x]?
|
1/26
|
27, 26
|
Unique
Factorization and Factor Rings
|
Homework
1: Section 23: 3, 9, 12, 21, 27, 28;
Section 27: 5, 8, 32
Quiz 4: Is 4x^2+12x+15 in Z[x] irreducible?
|
1/28
|
29
|
Extension
Fields
|
Quiz 5: What is
the property of ideals that is important in making R/I a ring when R is a
ring and I is an ideal?
|
1/31
|
29
|
Extension
Fields
|
Quiz 6: What is
an extension field of a field K?
|
2/2
|
|
|
Class canceled
due to snow closure
|
2/4
|
|
|
Class canceled
due to natural gas closure
|
2/7
|
30
|
Vector Spaces
|
Quiz 7: Give an
example of an algebraic extension of Q.
|
2/9
|
31
|
Algebraic
Extensions
|
Homework 2:
Section 29: 4, 8, 10, 14, 23, 25, 30, 31, 33; Section 30: 4, 9, 15
Quiz 8: Is a
spanning set of a vector space necessarily linearly independent?
|
2/11
|
31
|
Algebraic
Extensions
|
Quiz 9: Is a
finite field extension always algebraic?
|
2/14
|
31
|
Zorn’s Lemma
and Algebraic Closure
|
Quiz 10: If E is
an algebraic extension, what is \overline{F}E?
|
2/16
|
33
|
Finite Fields
|
Homework 3:
Section 31: 4, 10, 19, 23, 27, 29, 30, 32, 36
|
2/18
|
33
|
Finite Fields
|
Quiz 11: If [E:F]=n and F
has q elements, how many elements does E have?
|
2/21
|
34
|
Isomorphism
Theorems
|
Quiz 12: F_{p^n} is contstructed
inside what field?
|
2/23
|
35
|
Series of
Groups
|
Quiz 13: State
the 2nd Isomorphism Theorem.
|
2/25
|
|
Review
|
Quiz 14: Is {e}
< {e,f_1} <S_3 a subnormal series?
|
2/28
|
|
Midterm 1
|
|
3/2
|
35
|
Series of
groups
|
Homework 4:
Section 33: 2,8,9,12; Section 34: 4, 9; Section 35
4,5
|
3/4
|
35
|
Series of
groups
|
Quiz 16: Give
isomorphic refinements of the normal series (0) < (5) <Z_20 and (0)<(4)<(2)<Z_20.
|
3/7
|
35
|
Sylow Theorems
|
Quiz 17: Give
an example of a composition series.
|
3/9
|
36
|
Sylow Theorems
|
Homework 5:
Section 35: 8, 12, 13, 17, 18, 28, 29; Section 36: 2, 4, 6, 10, 12, 13, 16
Quiz 18: In a group
of order 18, what is the size of the Sylow
3-subgroup?
|
3/11
|
36
|
Sylow Theorems
|
Quiz 19: In a
group of order 50, what does the First Sylow
Theorem tell you about the orders of some of the subgroups?
|
3/21
|
37
|
Sylow Theorems
|
Quiz 20: What are the possible number of Sylow
5-subgroups in a group of order 20?
|
3/23
|
39
|
Free Groups
|
Homework 6:
Section 37: 3, 4, 6, 7; Section 39 1, 2, 4, 10
Quiz 21: Is a
group of order 21 necessarily abelian?
|
3/25
|
39
|
Free groups
|
Quiz 22: Is
there a homomorphism f from F[{x,y}]
to Z_6 with f(x)=2 and f(y)=3?
|
3/28
|
40, 48
|
Group
Presentations, Automorphisms of Fields
|
Quiz 23: If N
is a normal subgroup in G and H_i form a
composition series for G, what is a composition series for G/N?
|
3/30
|
|
Review
|
Homework 7:
Section 40 1, 2, 4, 8, 13
Quiz 24:
|
4/1
|
|
Midterm 2
|
|
4/4
|
48
|
Automorphisms of Fields
|
|
4/6
|
48
|
Automorphisms of Fields
|
Homework 8: Section
48 4, 8, 10, 12, 20, 29, 34, 36, 37
Quiz 26: What
is the fixed field of \sigma mapping Q(\sqrt{26}) to itself via \sigma(\sqrt{26})=-\sqrt{26}?
|
4/8
|
49
|
Isomorphism
Extension Theorem
|
Quiz 27:
|
4/11
|
49
|
Isomorphism
Extension Theorem
|
Quiz 28: What
is the partially ordered set that we used in proving the Isomorphism
extension theorem?
|
4/13
|
50
|
Splitting
Fields
|
Homework 9:
Section 49: 2, 4, 6, 8, 9, 11, 12, 13
Quiz 29: What
is the splitting field of x^p-x over Z_p?
|
4/15
|
50
|
Splitting
Fields
|
Quiz 30: Does
x^2-30 split over Q(\sqrt{2},
\sqrt{3}, \sqrt{5})
|
4/18
|
51
|
Separable
Extensions
|
Quiz 31: Tell
me something special about Z_p(y) and Z_p(y^p)
|
4/20
|
51
|
Separable
Extensions
|
Homework 10: Section 50: 2, 4, 10, 14, 15, 18, 20, 24;
Section 51 8, 9, 11, 15, 17, 18
Quiz 32: Is a
field with 32 elements perfect?
|
4/22
|
52
|
Totally
Inseparable Extensions
|
Quiz 33: What
is the primitive element theorem?
|
4/25
|
53
|
Galois Theory
|
Quiz 34: Give
an example of a normal field extension.
|
4/27
|
53
|
Galois Theory
|
Homework 11:
Section 52: 1, 2, 5, 7; Section 53 2, 8, 14, 15, 16, 17, 19, 23
|
4/29
|
56
|
Insolvability
of the Quintic
|
|
5/2
|
|
Review
|
|
5/4
|
|
Review
|
|
5/6
|
|
Review
|
|
5/13
|
|
Final exam
|
|