Math 4108 FInal Exam, Spring 2010
May 5, 2010
1. Dene the following terms
a. Euclidean Domain.
b. Constructible number.
c. Splitting eld.
d. Jordan-Holder Theorem.
e. Associative algebra.
2. Show that in a principal ideal domain, any nested chain of ideal
Homework on Derived Series, and Upper and Lower Central Series
April 12, 2010
1. We say that a subgroup H of a group G is characteristic if every automorphism : G G maps H H . Note that if G has only a single subgroup H of some order n, then H is characte
Solutions to Homework 4
May 1, 2010
1. We will prove this via induction: The rst term in the Upper Central
Series is cfw_e, which is clearly a characteristic group. Suppose, for proof by
mathematical induction, we have shown that
cfw_e = Z0 Z1 Zk
are all
Syllabus for Math 4108, Abstract Algebra
January 11, 2010
Instructor: Ernie Croot email: [email protected]/ecroot Please resist the urge to email me unless it is absolutely necessary. Course Webpage: www.math.gatech.edu/ecroot Click on the Math 4108
Math 4108 midterm 2 study sheet
April 20, 2010
Know the fact that if f (x) F [x], where F is a eld, and f (x) has a double root, then f (x) and f (x) have a common factor. Know that this is if and only if in elds of characteristic 0 (i.e. in ch. 0, f has
Math 4108 Study Sheet for Midterm 1
February 23, 2010
1. Know the books denition of prime elements, irreducible elements of polynomial rings. Know how to show that if p is a prime element of a PID, and p|ab, then p|a or p|b (on the exam, I will tell you w
Math 4108 midterm 2, Spring 2010
April 30, 2010
1. Dene the following terms a. Galois group of a eld K over a eld F (Also, what properties must K have in order for the Galois group to even exist?) b. Lower Central Series. c. Fundamental Theorem of Finitel
Midterm 1, Math 4108, Spring 2010
March 1, 2010
1. Dene the following terms. a. Algebraic extension. b. Unique Factorization Domain. c. Splitting eld. d. Eisensteins criterion. e. algebraic number. 2. Compute the degree of the extension F [x]/(x2 2) over
Final Exam Study Sheet
April 30, 2010
You will have 10 questions, and 3 hours to complete the exam. If you score is higher on the nal exam than you score up to and including the nal, then your course grade will be the nal exam grade (maximal grading poli
Pade Approximations and the Transcendence
of
Ernie Croot
March 9, 2007
1
Introduction
Lindemann proved the following theorem, which implies that is transcendental:
Theorem 1 Suppose that 1 , ., k are non-zero algebraic numbers, and that
1 , ., k are dist