MATH 601
Homework 3
Due September 27, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. (von Dycks theorem) Let X be a set and let R be a set of reduced words on X . Assume
that
MATH 601
DETAILED COURSE OUTLINE
Groups and Monoids
Groups and monoids, subgroups and submonoids, subgroup criterion, [JI, 1.1-1.5, pp.2647].
Symmetric, alternating, dihedral, matrix groups, free groups and presentations of groups,
[JI, 1.6, pp.48-51; 1
MATH 601
Homework 5
Due October 11, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. Do Exercise 7, p.100.
2. Do Exercise 8, p.100.
3. Do Exercise 6, p.103.
4. Do Exercise 8, p
Math 671 Homework #2 Solutions
17 #6 (a) Take any a A; then for any neighborhood U of a we have
U A = . This implies that U B = , since A B , and so
a B.
Or: Note that B is a closed set containing A, so since A is the
smallest closed set containing A, we
Math 671 Homework #1 Solutions
13, #7 The standard topology T1 is contained in (i.e. is coarser than) the
topologies T2 and T4 , and it contains (is ner than) T3 and T5 . All
of these inclusions are proper. There is a strict inclusion T2 T4 are
not compar
Math 671 Homework #4 Solutions
23, #5 If X has the discrete topology, then it is totally disconnected: any
subset with more than one point is disconnected, since it can be partitioned into two disjoint nonempty subsets, which are automatically
open, and s
Math 671 Homework #3 Solutions
20 #4 (a) Y = continuous; N = not continuous.
box unif prod
fN
N
Y
gN
Y
Y
hN
Y
Y
(b) Y = converges; N = does not converge.
box unif prod
wN
N
Y
xN
Y
Y
yN
Y
Y
zY
Y
Y
#5 The closure of R in the uniform topology is
S = cfw_a =
Math 671 Homework #5 Solutions
26,#3 Let X = K1 Kn , where each Ki is compact. Take an open
cover A of X . Its an open cover of Ki for all i, so we can nd nitely
many Ui,1 , . . . , Ui,ri A which cover Ki . Then the set of all Ui,j for
1 i n and 1 j ri is
Math 671 Homework #6 Solutions
29 #8 Let X be the one-point compactication of Z+ , with the added
point, and let Y = cfw_0 cfw_1/n | n Z+ . There is an obvious map
f : Y X given by f (1/n) = n and f (0) = . It is clearly a
bijection, so it is enough to sh
MATH 601
Homework 2
Due September 20, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. Do Exercise 1, p.53.
2. Do Exercise 1, p.63.
3. Do Exercise 1, p.70.
4. Let G be the subg
MATH 601
Homework 6
Due October 18, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. Do Exercise 6, p.146.
2. By Theorem 2.7, p.132 a polynomial f (x) of degree n > 0 with coec
MATH 601 SYLLABUS
Instructor: Maxim Arap
Course: MATH 601
Oce hours: Tuesday 9-11 AM
E-mail: marap@math.jhu.edu
Oce: 311 Krieger Hall
Course description
This is the rst semester of graduate algebra course. The highlights of this course are the
following g
MATH 601
Homework 7
Due October 25, 2012
1. Let A be a commutative ring, let S and T be two multiplicatively closed subsets of A,
and let U be the image of T under the localization map : A S 1 A. Show that the
rings (ST )1 A and U 1 (S 1 A) are isomorphic
MATH 601
Homework 8
Due November 1, 2012
1. Let V be a vector space over a eld F and let T : V V be a linear transformation.
(a) Let x be an indeterminate over F . Prove that the assignment x.v = T (v ) gives an
F [x]-module structure to V .
(b) Prove tha
MATH 601
Homework 9
Due November 8, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. Do Exercise 1, p.165.
2. Do Exercise 2, p.165.
3. Do Exercise 4, p.166.
4. Do Exercise 5, p
MATH 601
Homework 11
Due November 20, 2012
1. Let F be a eld and a(x) F [x] be a non-constant monic polynomial. Prove that the
characteristic polynomial of the companion matrix Ca(x) of a(x) is a(x).
2. Find all similarity classes of 6 6 matrices over Q w
MATH 601
Homework 4
Due October 4, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. Do Exercise 3, p.82.
2. Do Exercise 5, p.382.
3. Do Exercise 15, p.84.
4. Do Exercise 16, p.
MATH 601
Homework 10
Due November 15, 2012
1. Let R be a PID and let p R be a prime ideal with quotient eld F = R/p. Prove the
following statements:
(a) If M = Rn then M/pM
F n.
(b) If M = R/(a) for some a R\cfw_0 then M/pM
if p does not divide a.
F if p
MATH 601
Homework 1
Due September 13, 2012
The exercise numbers and page references are made to the rst volume of Jacobsons Basic
Algebra unless stated otherwise.
1. (a) Prove the subgroup criterion: Given a group G, H G is a subgroup if and only
if a1 b
MATH 601
Homework 12
Due December 6, 2012
1. Determine all possible Jordan canonical forms for a linear transformation with characteristic polynomial (x 2)3 (x 3)2 .
2. Let
5
41
A = 1 0 0 .
3 4 1
Find the matrix P such that P 1 AP is in Jordan canonical f
Math 671 Homework #7 Solutions
43, #4 Suppose X is complete and take a nested sequence A1 A2
A3 . . . of nonempty closed bounded sets with diam(An ) 0.
Form a sequence cfw_xn by letting xn be any point in An . Then xn
is Cauchy: for any > 0 there exists