Homework 1
Due Wednesday, April 10, at noon
Homework is due at noon on the following Wednesday. You are encouraged to work together with others,
but you must write up the solutions on your own.
All numbered exercises are from Dummit and Foote, third editi

Selected Solutions to
Loring W. Tus
An Introduction to Manifolds (2nd ed.)
Prepared by Richard G. Ligo
Chapter 1
Problem 1.1: Let g : R R be defined by
Z t
Z t
3
g(t) =
f (s)dt =
s1/3 dt = t4/3 .
4
0
0
Rx
Show that the function h(x) = 0 g(t)dt is C 2 but

Math 202 - Assignment 5
Authors: Yusuf Goren, Miguel-Angel Manrique and Rory Laster
Exercise 14.8.1.
Proof. The discriminant of x4 + 1 is D = 256 = 28 . We have x4 + 1 (x + 1)4 (mod 2). Let p be
an odd prime (so p - D), and suppose the irreducible factors

Problem 2.
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1. Show that the following rules constitute (left) group actions on the speci-
fied sets:
(a) Let F be a eld and F" - FHO the multiplicative group of nonzero
elements of F. Then F" acts on F via 9 - a - ga where g E F",a E F.
i: Leta E 44,91,92 E F". Then

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Universitext
Editorial Board
(North America):
S. Axler
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For other titles in this series, go to
www.springer.com/series/223
Loring W. Tu
An Introduction to Manifolds
Second Edition
Loring W. Tu
Department of Mathematics
Tufts University
Medford,

Bob Lutz
Math 437
HW # 2
9/23/13
Acknowledgments: I discussed these problems with Trevor Hyde.
Lee: 2-14, 3-4, 3-6, 3-8
(2-14) Let : R T2 be the curve of Example 4.20. Show that the image set pRq is dense in T2 .
Since t e2it is periodic with period 1, it

(January 14, 2009)
[14.1] Show that Q( 2) is normal over Q.
We must show that all imbeddings : Q( 2) Q to an algebraic closure of Q have the same image. Since
(by Eisenstein and Gauss) x2 2 is irreducible in Q[x], it is the minimal polynomial for any squa

Adam Cross
Solutions for Lees Smooth Manifolds
Lee 5-1
Let E be a vector bundle over a topological space M . Then the projection map : E M is a homotopy
equivalence.
Proof. Let us begin by reminding ourselves that a homotopy equivalence is a map f : X Y s

Lie Bracket of two vector fields
1. The definition of a Lie bracket.
Let M be an n-manifold. Recall that if X and Y are smooth vector fields
on M then X and Y are 1-st order differential operators on smooth functions
M R. Thus, for a smooth function f : M

M a 5b HOMEWORK 7 SOLUTION
WINTER 10
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote.
Page 356,3. Any finite abelian group M is a Zmodule. Suppose
|M | = n, then nm = 0 for all m M . Therefore M is a torsion
Zmo

M a 5b HOMEWORK 6 SOLUTION
WINTER 10
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote.
Page 344,15. Suppose M is a group of order n, then for any m M ,
nm = 0. Now 0 = n1 (nm) = ( n1 n)m = 1 m 6= m. Therefore M c

MATH 332: HOMEWORK 4
Problem 1. Let 3 t1, 2, 3u and let S3 act on 32 3 3 via pi, jq
ppiq, pjqq.
(a) Find the orbits of S3 on 32 .
(b) For each P S3 find the cocycle decomposition of under this action.
(I.e., the action affords a permutation representatio

MATH 332: HOMEWORK 3
Problem 1. Prove that if H and K are finite subgroups of G whose orders
are relatively prime, then H K = 1.
Problem 2. Use Lagranges Theorem in the multiplicative group (Z/pZ) to
prove Fermats little theorem: if p is prime, then ap a

M a 5b HOMEWORK 3 SOLUTION
WINTER 10
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote.
Page 267,1. Note (1e)2 = 12e+e2 = 1e. Take r, s R, then re+
se = (r + s)e, (re) = (r)e,r(se) = (rs)e, (se)r = (sr)e. Therefor

MATH 332: HOMEWORK 11
Exercise 1. Compute
HomZ pZcfw_65Z bZ Zcfw_91Z, Zcfw_131Zq
as an abelian group.
Exercise 2. Find, with proof, the number of finitely generated abelian groups
of order 100. Do the same for finitely generated abelian groups of order 57

MATH 332: HOMEWORK 9
Exercise 1. Let ppx, y, zq 2x2 3xy 3 z ` 4y 2 z 5 and qpx, y, zq 7x2 `
5x2 y 3 z 4 3x2 z 3 be polynomials in Zrx, y, zs. Let ppx, y, zq and qpx, y, zq denote the images of ppxq and qpxq in pZcfw_3Zqrxs under the canonical reduction
ma

MATH 332: HOMEWORK 8
Exercise 1. For each of the following pairs of integers a and b, use the Euclidean algorithm to determine their greatest common divisor d and write
d as a linear combination ax ` by of a and b.
(a) a 20, b 13.
(b) a 69, b 372.
(c) a 9

M a 5b HOMEWORK 2 SOLUTION
WINTER 10
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote.
Page 249,18.(a)I J is nonempty since it contains 0. Take any a, b
I J, then clearly a b I J. For any r R, ra, ar I J.
Theref

MATH 332: HOMEWORK 5
Exercise 1. Exhibit all Sylow 2-subgroups and Sylow 3-subgroups of D12
and S3 S3 .
Exercise 2. Exhibit two distinct Sylow 2-subgroups of S5 and an element of
S5 that conjugates one into the other.
Exercise 3. Exhibit all Sylow 3-subgr

M a 5b HOMEWORK 5 SOLUTION
WINTER 10
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote.
Page 306,2. Let h(x) = f (x)g(x) Z[x]. By Gauss Lemma, theres a
factorization h(x) = a(x)b(x) in Z[x] with a(x) = rf (x), b(x

MATH 332: HOMEWORK 7
Exercise 1. Let R be a ring with 1. Prove that p1q2 1 P R and that if u is
a unit in R, then u is also a unit in R.
Exercise 2. Which of the following are subrings of Q:
(a) the set of rational numbers with odd denominators (when writ

MATH 332: HOMEWORK 6
Exercise 1. Construct your own handcrafted, artisanal semi-direct product:
carefully select groups H and K from a small batch producer in the Cascades. Choose an explicit nontrivial homomorphism : K AutpHq.
Write the product on H K ex