3520.001, Spring 2004
Abstract Algebra II
the University of North Texas.
Instructor: Charles Conley, GAB 419, (940) 565-3326
Office hours: MW 1:30-3:00, F 1:30-2:30
Class meets: MWF 12:00-12:50, Lang 315
Exams, homework, and grading: There will be two 100 point midterms and a comprehensive 175 point final. There will be thirteen homeworks (totalling 125 points), due at the beginning of class each Friday excepting exam weeks. 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 Abstract Algebra I, 3510, or its equivalent. You will need to know elementary group theory and elementary ring theory: familiarity with groups of small order and the ring Z_n of integers modulo n will be sufficient. A little knowledge of linear algebra at the level of 2700 (e.g., bases of vector spaces) will also help, but we will cover this material in the course.
Topics: The course will focus on fields, field
extensions, and Galois theory. We will begin with polynomial
rings and ideals, followed by vector spaces over arbitrary fields.
Then we will learn how to construct extension fields from
irreducible polynomials, and how to study such fields using Galois
theory. Along the way we will cover as many of the following
topics as possible.
The Hamiltonians: These form a skew field structure on R^4, the simplest skew field, in fact, the only skew field on R^n for any n. The unit Hamiltonian sphere is a multiplicative group isomorphic to a group you may have heard of: the matrix group SU_2.
Straightedge and Compass Constructions: Several famous classical problems can be solved easily using the theory of field extensions. For example, we will prove that it is impossible to trisect angles, square the circle, or double the cube. We will also consider the construction of regular polygons: for prime-gons, this is possible only for Fermat primes such as 3, 5, 17, 257, or 65537 (it is not known if there are any others).
Finite Fields: It is a beautiful fact that there is exactly one field of size p^n for each prime p and positive integer n, and these are all the finite fields. We will prove this and discuss the discrete logarithm, which is used in the encryption of digital signatures.
Polynomial Equations: It is an ancient problem to generalize the quadratic formula to polynomials of higher degree. Galois theory, coupled with the structure of the symmetric groups S_3 and S_4, easily yield the formulae for the solution of the general cubic and quartic equations. The same methods, together with the fact that the only normal subgroup of S_5 is A_5, prove that the general quintic cannot be solved.
It is the responsibility of students with certified disabilities to provide the instructor with appropriate documentation from the Dean of Students Office.
HOMEWORK 11, due the day of the final exam.
Starred problems will not be graded, but
problems similar to them may be on the final. In problems
involving Q bar, the algebraic
closure of Q, replace it by the complex numbers C. In 9.1.30,
F bar by the splitting field of the irreducible polynomial of
alpha over F.
HOMEWORK 10, due Wednesday, April 21:
HOMEWORK 9, due Friday, April 9:
Section 8.3: 7-13, 25-28
Section 8.4: Two supplemental problems:
A) (i) Find irr_Q(exp(2i pi /7))
(ii) Find irr_Q(cosine(2 pi/7))
(iii) Prove that the regular septagon is not constructible
B) (i) Prove that if 2^n-1 is prime, then n
(ii) Prove that if 2^n+1 is an odd prime, then n is 2^k for some natural number k
HOMEWORK 8, due Wednesday, Mar. 29, the day
of Exam 2:
Section 8.1: 25-26
Section 8.2: 5-10
Section 8.3: 1-6, 20-23
HOMEWORK 7, due Friday, Mar. 12 (Note: due
postponed by one week.) (Also, Problem 16
been replaced by Problem 10):
HOMEWORK 5, due Wednesday, Feb. 18 (This
problem set is worth only 5 points and is due the day of Exam 1):
HOMEWORK 4, due Wednesday, Feb. 11 (15
points) (Note the increased value and extended due date!):
HOMEWORK 3, due Friday, Jan. 30 (10 points):
Section 5.6: 2, 4-10, 12, 14, 16, 18, 20, 32-35
HOMEWORK 2, due Friday, Jan. 23 (10 points):
Section 5.5: 3-6, 10-12, 15-17, 24, 25, 27
HOMEWORK 1, due Friday, Jan. 16 (5 points): See handwritten problems