Name:
Problem Set 1
Math 541, Fall 2014
Due: Thursday, September 25
Complete the following items, staple this page to the front of your work, and turn your assignment
in class on Thursday, September 25.
Modular arithmetic
1. Determine the last digit of 34
FINAL, MATH 541 - DECEMBER 17, 2011
Nigel Boston
1. Let be a mapping from set S to set T .
(a) Define what it means for to be 1-1. Define what it means for to be onto.
(b) , (c) Give an example of a set S and a mapping : S S that is 1-1 and
not onto (resp
LAST HOMEWORK, MATH 541
Nigel Boston
For full credit explanations are necessary.
1. (a) Compute the greatest common divisor d of 105 and 144,
(b) Find integers x and y such that d = 105x + 144y.
2. (a) List all the subgroups of the group (Z12 , +).
(b) Do
2ND MIDTERM, MATH 541 - NOVEMBER 10, 2011
Nigel Boston
1. Let G and H be groups.
(a) Define homomorphism from G to H. Define its image (G).
(b) Show that if G is abelian and is onto, then H is abelian.
2. Let G be the group of all continuous real-valued f
UW-Madison
Department of Mathematics
Math 541
Assignment 1
Spring 2017
The homework is due Friday, Feb. 03 in the beginning of the lecture. Please staple your homework and dont
forger to write your name and student number on the first page of your homewor
UW-Madison
Department of Mathematics
Math 541
Assignment 2
Spring 2017
The homework is due Friday, Feb. 10 in the beginning of the lecture. Please staple your homework and dont
forger to write your name and student number on the first page of your homewor
Assignment 12 Parts 1 & 2 Math 541
Due Friday, Dec. 14, at the beginning of class
The following problems from the textbook:
Section 3.2: 2, 3, 9
Section 3.3: 1, 5
(1) In this question, R is a commutative ring with identity. This question continues the
ide
Assignment 7 Parts 1, 2, & 3 Math 541
Due Friday, Nov. 2, at the beginning of class
The following
Section 2.4:
Section 2.6:
Section 2.7:
problems from the textbook:
13, 14, 18, 24, 31 and 29
7, 8, 11, 12, 13 and 1, 3, 4
2, 3
(1) Let Q be the quaternion gr
Assignment 8 Final version Parts 1, 2, & 3 Math 541
Due Monday, Nov. 12, at the beginning of class
(1) Recall from assignment 6 question (4) that, for h1 , h2 G, we say that h2 is conjugate to h1 if there is g G such that
h2 = g 1 h1 g.
(The map from G to
1ST MIDTERM, MATH 541 - OCTOBER 11, 2011
Nigel Boston
Answer all five questions below, writing your answers on the paper provided.
You may use the extra sheet at the end for scrap (and remove it). Each question is
worth the same amount. Show your working.
Name:
Problem Set 5
Math 541, Fall 2014
Due: Thursday, October 9
Euclidean algorithm for polynomials
1. Show by example that unique factorization fails in R[x] when R is not an integral domain. For
instance, consider x2 + x + 8 Z10 [x].
Roots of polynomia
Name:
Problem Set 1
Math 541, Fall 2014
Due: Thursday, September 11
Complete the following items, staple this page to the front of your work, and turn your assignment
at the beginning of class on Thursday, September 11
1. Carefully read the entire course
Name:
Problem Set 2
Math 541, Fall 2014
Due: Thursday, September 18
Complete the following items, staple this page to the front of your work, and turn your assignment
in class on Thursday, September 18.
Properties of the integers
1. Prove that the square
Name:
Problem Set 10
Math 541, Fall 2014
Due: Thursday, December 4
Group homomorphisms and isomorphisms
1. Show that : R C given by (t) = cos(2t) + i sin(2t) is a homomorphism. Show that
Z is the kernel of and the unit circle in the complex plane is the i
Name:
Problem Set 6
Math 541, Fall 2014
Due: Thursday, October 23
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 ideals
Name:
Problem Set 9
Math 541, Fall 2014
Due: Tuesday, November 18
Groups
1. Let G be a group and fix a G. Prove that Ca = cfw_x G | ax = xa is itself a group, called
the centralizer of a.
Cyclic groups
2.
a. List all of the generators of Z20 .
b. List the
Name:
Problem Set 4
Math 541, Fall 2014
Due: Thursday, October 2
Rings, domains, and fields
1. Determine if the set R = cfw_a + b 3 3 | a, b Q is a ring with respect to the usual operations
of addition and multiplication. If so, is it also a field?
2. Cha
Name:
Problem Set 7
Math 541, Fall 2014
Due: Thursday, October 30
Quotient rings
1. Give the addition and multiplication tables of Z2 [x]/hx3 + x + 1i.
2. Let R and S be commutative rings and : R S be a ring homomorphism.
a. Given an ideal J S, define 1 (
Name:
Problem Set 8
Math 541, Fall 2014
Due: Thursday, November 6
Vector spaces and field extensions
1. Prove
that the real numbers 1 and
and 5.
3 are linearly independent over Q. Do the same for 1, 3,
2. Give a basis for each of the given vector spaces o
Name:
Problem Set 11
Math 541, Fall 2014
Due: Thursday, December 11
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 finite group of order n. Let Perm(G) denote
Assignment 9 Parts 1, 2, & 3 Math 541
Due Monday, Nov. 26, at the beginning of class
The following problems from the textbook:
Section 4.1: 14, 15, 16
Section 4.2: 1, 3, 81
(1) The goal of this problem is to classify groups of order 8.
(a) There are three
Assignment 11 Parts 1 & 2
The following problems from the textbook:
Section 4.5: 8, 10, 12
(1) This will show up in class on Monday. Let : R R be a ring homomorphism
and let I be an ideal in R. Give an example where (I ) is not an ideal in R . But
show th
Math 541 Spring 2013
HW#5 Solutions Orthogonal Symmetries of the n-Regular
Polygon, Subgroups of Z; Product, Cyclic Groups
4/6/13
Remark. Answers should be written in the following format:
i) Statement and/or Result.
ii) Main points that will appear in yo
Math 541 Spring 2013
Solutions
HW7, Normal Subgroups, Lagrange Theorem, Groups
s
of Prime Order
April 29 2013
Remark. Answers should be written in the following format:
0) Statement and/or Result.
i) Main points that will appear in your explanation or pro
Math 541 Spring 2013
Solutions HW8 Orbits,Cosets, Lagrange Theorem, Fermat
s
s
Little Theorem
04/29/13
Remark. Answers should be written in the following format:
0) Statement and/or Result.
i) Main points that will appear in your explanation or proof or c
Math 541 Spring 2013
Homework#4 Solutions
03/16/13
1. The orthogonal group. Denote by h ; i the standard inner product on V = R2 : Consider
the set O = fA 2 M at(2; R); such that h Au; Av i = h u; v i for every u; v 2 R2 g; i.e.,
the set of all 2 2 real m
Math 541 Spring 2013
Preparation for the Final Test
Remarks
Answer all the questions below.
A denition is just a denition there is no need to justify it. Just write it down.
Unless it a denition, answers should be written in the following format:
s
Write
Math 541 Spring 2013
HW3 Answers Rotational Symmetries, Subgroups of GL2(R)
03/04/2013
Consider the group (R; ; I ) of rotations of the plane R2 of all matrices of the form
r=
with
cos( )
sin( )
sin( )
,
cos( )
denotes multiplications of matrices, and I =
Math 541 Spring 2013
Preparation for the Mid-Term Test
Remarks.
Answer ALL the questions (a) and (b) below, and only one of the (c)
s
s
s.
Denition (subsections (a) is just a denition and there is no need to justify it. So
just write it down.
Answers to s