A. Miller
M541
Exam 2 Answers
Fall 99
1. Problem-Points: 1-5 2-5 3-5 4-4 5-4 6-4 7-9 8-3 9-2 10-5 11-6 12-5 (a) f : X Y is one-to-one iff x, y X f (x) = f (y) x = y (b) f : X Y is onto iff y Y x X f (x) = y (c) g is the inverse of f iff g : Y X and x X y
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15.Is the sequence 3, 12, 36, . a geometric sequence? Explain.Type your answer
below.
NO! A geometric sequence goes from one term to the next by always multiplying
(or dividing) by the same value. So 1, 2, 4, 8, 16,. and 81, 27, 9, 3, 1,
1/3,. are geo
A. Miller
M541
Review for Exam 2
Fall 99
1. Dene f : X Y is one-to-one. Dene f : X Y is onto. Dene g is the inverse of f . 2. Prove f is 1-1 onto i f has an inverse. 3. Dene transposition, n-cycle, disjoint cycles, parity, sign, crossing pair, crossing nu
A. Miller
M542
An Example
Spring 2000
Theorem. There exists a eld F and , in some extension eld of F such that [F (, ) : F ] < but there is no F [, ] such that F (, ) = F (). This is similar to the example on page 354 of Gallians book. Let F = Z2 (s, t) w
A. Miller
M541
Exam 1 Answers
Fall 99
1. (4points) Suppose p, n, m are integers. (a) n divides m iff k Z m = kn. (b) p is a prime number iff p 2 and there exists no k Z with 1 < k < n and k divides p. (c) gcd(n, m) = d iff d is the largest integer which d
A. Miller
M542
Final Exam
Spring 2000
The Final Exam is in our usual classroom (B203 Van Vleck) at 7:25pm on Saturday May 13. It consists of approximately six proofs from the material below which I will write on the blackboard. A copy of this document wil
A. Miller
M542
Galois Theory
Spring 2000
For the material on Galois theory we will be assuming that the elds all have
characteristic zero. When we get to solvability by radicals we will assume that all
elds are subelds of the complex numbers C.
1
Review
T
A. Miller
M340
April 97
edited Jan 2000 for M542
Vector Spaces
A vector space, V , is a set with two operations, vector addition (written
u + v) and scalar multiplication (written av). The elements of V will be
denoted using u, v, w, etc. The formula u V
A. Miller
M542
Spring 2000
1
Linear Transformations
In this section we consider only nite dimensional vector spaces V or W over an
arbitrary eld F.
Theorem 1.1 Every linear transformation L : Fn Fm is determined by an m n
matrix A:
L(X) = AX
for every X F
A. Miller
M542
Review for midterm
Spring 2000
1. Lemma. Suppose G be a nite abelian group such that |G| = mn where m and n are relatively prime. Let H = cfw_x G : xn = e and K = cfw_x G : xm = e. Then H and K are subgroups of G and G = H K H K. 2. Lemma.