MAT 150A
University of California
Fall 2015
Homework 2
due October 9, 2015 in class
Read: Artin Chapters 2.3, 2.4, 2.5
1. Artin 2.1.1 (pg. 69)
Let S be a set. Prove that the law of composition defined by ab = a for all
a, b S is associative. For which set
MAT 150B
University of California
Winter 2014
Homework 3 Solutions
1. Find a subgroup of GL2 (R) which is isomorphic to C .
Solution: Comparing matrix multiplication with multiplication of complex
numbers, we are lead to the subset of GL2 (R) consisting o
Midterm Exam: 150B
February 14, 2014
This is a 50 minute, closed book exam. No notes, books or calculators are allowed. Clear
exposition will be given credit. Please write all answers on these sheets.
This exam consists of 4 problems.
Good luck!
MAT 150B
University of California
Winter 2014
Homework Solutions 5
1. Let be a representation of a group G. Show that det is a onedimensional representation.
Solution: Let
: G GL(V )
g g
be a homomorphism and let det : GL(V ) F = GL1 (F) be the determina
MAT 150B
University of California
Winter 2014
Homework Solutions 7
1. Prove the following identities in an arbitrary ring R from the axioms.
(a) 0a = 0
(b) a = (1)a
(c) (a)b = (ab)
Solution:
(a) We have
0a + a = 0a + 1a = (0 + 1)a = 1a = a.
This means 0a
MAT 150B
University of California
Winter 2014
Homework Solutions 8
1. Prove or disprove: If an ideal I contains a unit, then it is the unit ideal.
Solution: Let I be an ideal of the ring R. Let a R be a unit such that
a I. By the property of a unit, there
MAT 150B
University of California
Winter 2014
Homework Solutions 6
1. Determine the character table for D6 .
Solution: We use the generators x and y of D6 with the usual relations
x6 = 1,
y 2 = 1,
xy = yx5 .
The conjugacy classes of D6 are
cfw_1, cfw_x3 ,
150C HOMEWORK 1 SOLUTIONS
NATHANIEL GALLUP
(1) Artin 12.2.3
Claim 1. The polynomial x2 2 has no roots when considered as a polynomial in Z/8[x].
Proof. The roots of the polynomial x2 2 are exactly the square roots of 2 in Z/8, i.e. the elements of
Z/8 tha
150C HOMEWORK 3 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1. Let a, b be elements of a eld F with a = 0. Then f (x) F [x] is irreducible if and only if
f (ax + b) is irreducible.
Proof. First assume that f (x) is irreducible, and suppose for contradiction that
150C HOMEWORK 4 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1.
(a) Let V be an abelian group. If V has a Q-module structure, then this structure is uniquely determined.
(b) No nonzero nite abelian group has a Q-module structure.
Proof.
(a) Suppose we already hav
150C HOMEWORK 5 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1. Let I be an ideal of a ring R, and suppose that R/I is a free R-module. Then I = 0. (If we
consider the 0 module to be free with empty basis then I can equal R as well).
Proof. Suppose for contradict
150C HOMEWORK 2 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1. Division with remainder is not unique in the integers, which is a Euclidean domain with size
function | |.
Proof. Suppose we want to divide 1 Z by 2 Z with remainder. There are two possible options.
150C HOMEWORK 6 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1. The matrix
1
1
1
A = 1 1 1
1
1
1
is idempotent, and has Jordan Form
0 0 0
0 0 0
0 0 1
Proof. First, direct computation shows that A2 = A, hence by denition, A is idempotent.
Now the characteristic p
150C HOMEWORK 7 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1. If is a root of the polynomial x3 3x + 4, then the inverse of 2 + + 1 in Q() is 3.492
5/49 + 17/49.
Proof. We are looking for some element c2 + b + a with a, b, c Q such that
(c2 + b + a)(2 + + 1) =
150C HOMEWORK 8 SOLUTIONS
NATHANIEL GALLUP
(1)
Claim 1. The polynomial xp x (for p prime) has multiple roots in a eld extension of F if and only if
the characteristic of F is 2 and p is an odd prime.
Proof. The polynomial f (x) = xp x = x(xp1 1) has a mul
A GENTLE INTRODUCTION TO
ABSTRACT ALGEBRA
by
B.A. Sethuraman
California State University Northridge
ii
Copyright 2012 B.A. Sethuraman.
Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation Lice
MAT 150B
University of California
Winter 2014
Homework Solutions 4
1. Let Q U2 (C) and = det Q. Show that if
= 1, and det(Q) = 1.
is a square root of , then
Solution: We have 2 = . Also, since Q U2 (C), we have Q = Q1 .
Therefore 1 = det Q det Q = . So w
MAT 150B
University of California
Winter 2014
Homework Solutions 1
1. (cf. Artin 8.1.1)
(a) Prove that every real square matrix is the sum of a symmetric matrix
and a skew-symmetric matrix (At = A) in exactly one way.
Solution: Let A be an arbitrary real
MAT 150B
University of California
Winter 2014
Homework Solutions 2
1. Let G = GLn (R) and let S = Mn (R) be the set of all n n matrices over
R. Show that the map G S S given by
(P, A) (P t )1 AP 1
denes an action of G on S.
Solution: We have
(I, A) = (I t
MAT 150A
University of California
Fall 2015
Homework 3
due October 16, 2015 in class
Read: Artin Ch. 2.5, 2.6
1. Artin 2.6.8 (pg. 72)
Prove that the map f : A 7 (At )1 is an automorphism of GLn (R).
2. Prove that the kernel and image of a homomorphism are
MAT 150A
University of California
Fall 2015
Homework 6
due November 13, 2015 in class
Read: Artin 5.1, 6.1, 6.2
1. Artin 5.1.1(a) pg. 150
What is the matrix of the three-dimensional rotation through the angle
about the axis e2 ?
2. (a) Prove that On and
MAT 150A
University of California
Fall 2015
Homework 1
due October 2, 2015 in class
We will use Artins numbering system so that Artin 1.4.2 means
Chapter 1, Section 4, Problem 2.
You are expected to hand in all problems. The reader will pick
several probl
MAT 150A
University of California
Fall 2015
Homework 4
due October 23, 2015 in class
Read: Artin 2.7-2.10
1. Let S be a set of groups. Prove that the relation G H if G is
isomorphic to H is an equivalence relation on S.
2. Let H be a subgroup of a group G
MAT 150A
University of California
Fall 2015
Homework 5
due November 6, 2015 in class
Read: Artin 2.11-2.12
(1) Artin 2.11.6 (pg. 74)
Let G be a group containing normal subgroups of orders 3 and
5, respectively. Prove that G contains an element of order 15
Math 150B Discussion Section 5
TA: Patrick Tam
February 6, 2014
Tetrahedral Group) Let T denote the tetrahedral group. Let yi T denote the rotation by around
an edge, and let x T denote rotation by 2/3 around the front vertex. The matrices representing th
Discussion Section Day 8
February 28, 2014
10.5.5) Prove that the one-dimensional characters of a group G form a group under multiplication of
functions. This group is called the character group of G, and is often denoted by G. Prove that if G is
abelian,
Math 150B Discussion Day 4
TA: Patrick Tam
January 30, 2014
9.3.2) Prove that U2 is homeomorphic to the product S3 S1 .
The map in proposition 9.3.2 on page 266 gives us a homeomorphism from SU2 to S3 . We note that
SU2 U1 U2
=
with the homeomorphism give
Math 150B Discussion Day 7
TA: Patrick Tam
February 20, 2014
10.4.2) A nonabelian group G has order 55. Determine its class equation and the dimensions of its
irreducible characters.
By the Sylow theorems, there is 1 normal Sylow-11 group and either 1 or
Math 150B Discussion Day 6
Patrick Tam
February 13, 2014
2.2) Consider the standard two-dimensional representation of the dihedral group Dn . For which n is this
an irreducible complex representation?
The standard representation of Dn is given by
cos
sin
Math 150B Discussion Section 10
March 14, 2014
11.8.2 (b) R[x]/(x2 ) has (x + (x2 ) as its unique maximal ideal.
Proof: Let (x): R[x] R[x]/(x2 ) be the quotient homomorphism, which maps g(x) to g(x) + (x2 ).
Let I R[x]/(x2 ) be a non-zero proper ideal (di