Name:
Problem Set 6
Math 4281, Fall 2013
Due: Friday, October 18
Read Sections 21.1 (stop after Example 1), 21.2 (stop after Example 11), and 16.3, as well as
Theorem 17.3 in your textbook.
1. Prove that Q( 2, i) = Q( 2 + i), but Q( 2i)
Q( 2, i).
2. Find
Name:
Problem Set 9
Math 4281, Spring 2014
Due: Wednesday, April 2
Ring isomorphisms
1. Establish the following isomorphisms by using the Fundamental Homomorphism Theorem:
a. R[x]/ x2 + 6 C
=
2
b. Q[x]/ x + x + 1 Q( 3i)
=
c. Z3 Z4 Z12
=
2. Let F be a eld,
Name:
Problem Set 1
Math 4281, Fall 2013
Due: Friday, September 13
Review Sections 1.1, 1.2 thru p. 14, 2.1 in your textbook, and read Section 2.2.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at th
Name:
Problem Set 2
Math 4281, Fall 2013
Due: Friday, September 20
Read Sections 3.1 (thru top of p. 40 only), 6.3 (skip Theorems 6.11 and 6.12), 16.5, and 1.2 in
your textbook.
Complete the following items, staple this page to the front of your work, and
Irena Swanson
Reed College
Spring 2014
Table of contents
Preface
7
The briefest overview, motivation, notation
9
Chapter 1: How we will do mathematics
13
Section 1.1: Statements and proof methods
13
Section 1.2: Statements with quantiers
24
Section 1.3:
Name:
Problem Set 8
Math 4281, Spring 2014
Due: Wednesday, March 26
Quotient rings
1. Prove that if F is a eld and f (x) F [x] is not irreducible, then F [x]/ f (x) contains zero
divisors.
2. Give the addition and multiplication tables of Z2 [x]/ x3 + x +
Name:
Problem Set 7
Math 4281, Spring 2014
Due: Wednesday, March 12
Ring homomorphisms and ideals
1. Find all ring homomorphisms:
a. : Z2 Z
b. : Z2 Z6
c. : Z6 Z2
2. Prove that if p is prime and : Zp Zp , (a) = ap , is a ring homomorphism.
3. Find all idea
Name:
Problem Set 5
Math 4281, Spring 2014
Due: Wednesday, February 26
The complex numbers
1.
a. Evaluate (4 5i) (4i 4).
b. Convert 5cis
9
4
to the form a + bi.
c. Change 2 + 2i to polar coordinates.
d. Calculate (i)10 .
e. Calculate
1i 4
.
2
2. Find the
Name:
Problem Set 6
Math 4281, Spring 2014
Due: Wednesday, March 5
Roots of polynomials
1. Prove that Q( 2, i) = Q( 2 + i), but Q( 2i)
Q( 2, i).
2. Find the splitting eld for the following polynomials in Q[x]:
a. f (x) = x6 1
b. f (x) = x4 10x2 + 1
(Hint:
Name:
Problem Set 3
Math 4281, Spring 2014
Due: Wednesday, February 12
Complete the following items, staple this page to the front of your work, and turn your assignment
in class on Wednesday, February 12.
Division and Euclidean algorithms
1. You have at
Name:
Problem Set 4
Math 4281, Spring 2014
Due: Wednesday, February 19
Equivalence relations
1. Dene a relation on R as follows: x y if and only if x y is an integer. Prove that is an
equivalence relation and describe the set of equivalence classes.
2. Gi
Name:
Problem Set 10
Math 4281, Spring 2014
Due: Wednesday, April 9
Groups
1. Which of the following are groups?
a. cfw_0, 2, 4, 6 Z10 , with operation addition
b. cfw_z C | |z| = 1, with operation multiplication
c. cfw_x Q | 0 < x 1, with operation multi
Name:
Problem Set 11
Math 4281, Spring 2014
Due: Wednesday, April 16
Permutation groups
1. Given the permutations = (1 2 4), = (1 3)(2 4) S4 , compute the following elements:
a. 1 b. c. d. 2 e. 2 f. 1 g. 1
2.
a. Prove that a k-cycle in Sn is an element of
Math 3283W: Sequences, Series & Foundations
Fall 2013 Course Syllabus
Math 3283W meets MWF at 11:15am-12:05pm in STSS 230. You also have sessions with your Teaching Assistants
on Tu/Th at either 10:10 or 11:15.
Instructor. Prof. Jonathan Rogness
Oce: Vinc
Math 4281
Instructor: Jessica Striker
23A Worksheet
April 26, 2013
Name:
Idea. Create new elds, called eld extensions, by adjoining new numbers to existing elds.
Example. Dene i as the solution to x2 + 1 = 0. Then R[i] is a new eld (that we usually
call C
Math 4281
Midterm Exam 2
March 4, 2013
Instructor: Jessica Striker
Name:
Instructions: You should write your solutions in the blue books. Show all of your work and
clearly explain your answers. No books or written notes are allowed during the exam, but
yo
Math 4281
Midterm Exam 3
April 19, 2013
Instructor: Jessica Striker
Name:
Instructions: You should write your solutions in the blue books. Show all of your work and
clearly explain your answers. No books or written notes are allowed during the exam, but
y
Math 4281
Instructor: Jessica Striker
17A Worksheet
April 24, 2013
Name:
Idea. We can do everything we did in Z/mZ with polynomials!
Denition. Let F be a eld and m a polynomial in F [x]. For f, g F [x], f g mod m if
m divides f g, that is, if f = g + hm f
Math 4281
Instructor: Jessica Striker
23BC Worksheet
April 29, 2013
Name:
Idea. Congruence modulo a polynomial yields new rings and elds.
Denition. Let F be a eld and m(x) a polynomial with coecients in F . The set of
congruence classes of polynomials mod
Math 4281
Midterm Exam 1
February 11, 2013
Instructor: Jessica Striker
Name:
Instructions: You should write your solutions in the blue books. Show all of your work and
clearly explain your answers. No books or written notes are allowed during the exam, bu
Name:
Problem Set 13
Math 4281, Spring 2014
Due: Wednesday, May 7
1. In this exercise, you will prove Cayleys Theorem, which says that every group is isomorphic
to a subgroup of a permutation group.
Let G be a nite group of order n. Let Perm(G) denote the
Name:
Problem Set 12
Math 4281, Spring 2014
Due: Wednesday, April 23
Cosets
1. Suppose G = c is a cyclic group of order 8. List the cosets of c4 .
Normal subgroups and quotient groups
2. Show that H is a normal subgroup of G, but K is not a normal subgrou
Name:
Problem Set 2
Math 4281, Spring 2014
Due: Wednesday, February 5
Complete the following items, staple this page to the front of your work, and turn your assignment
in class on Wednesday, February 5.
Properties of the integers
1. Prove that the square
Name:
Problem Set 1
Math 4281, Spring 2014
Due: Wednesday, January 29
Complete the following items, staple this page to the front of your work, and turn your assignment
in class on Wednesday, January 29.
1. Carefully read the entire course website. Send a
Mathematics 4281: solutions to Problem Set 5
(14.24) By Theorem 9.20, the order of An is exactly half the order of Sn ,
so An is an index-2 subgroup of Sn , and therefore is normal (by problem 10.39
from last weeks homework). The quotient group Sn /An has
Mathematics 4281: solutions to graded problems from PS 4
(9.39) (1, 2, . . . , n)r (1, 2)(1, 2, . . . , n)nr = (1, 2) when r = 0, (2, 3) when
r = 1, and in general, (r + 1, r + 2) for r = 0, 1, . . . , n 2. Finally, if r = n 1,
we get (n, 1). To see this,
Mathematics 4281: solutions to graded problems from PS 3
(6.18) The greatest common divisor of 42 and 30 is 6, so by Theorem 6.14,
42
the order of 30 in Z42 is gcd(42,30) = 42 = 7.
6
(6.36, 6.37) Every innite cyclic group is isomorphic to Z, which has exa
Mathematics 4281: solutions to graded problems from PS 2
(5.20) We have the set-theoretic containments 12Z 6Z 3Z Z R, all
of which are groups under addition with identity element 0, so we have subgroup
relations G2 G8 G7 G1 G4 . On the other hand, we have
Mathematics 4281: solutions to graded problems from PS 1
(2.1) b d = e, c c = b, and [(a c) e] a = [c e] a = a a = a.
(2.9) is both commutative and associative. Suppose a, b, c Q. Then
a b = ab = ba = b a, so is commutative, and a (b c) = a bc = a(bc/2) =
Mathematics 4281: Final Exam
Friday, 9 August 2013
You have two hours to complete this exam. No books, notes, calculators, or
cell phones are allowed. Unless otherwise indicated, all answers require justication. Please indicate clearly, in each section, w