MTH 581-582 Introduction to Abstract Algebra

276 Pages·2010·2.227 MB·English

by  D. S. Malik, John N. Mordeson, M. K. Sen

MTH 581-582
Introduction to Abstract Algebra
D. S. Malik
Creighton University
John N. Mordeson
Creighton University
M.K. Sen
Calcutta University
COPYRIGHT c 2007
°
Department of Mathematics
c2007byD.S.Malik. Allrightsreserved. Nopartofthisbookmaybereproduced,storedinaretrievalsystem,
°
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Printed in the United States of America
This document was produced with Scientific Word.
iii
iv
Preface
This book is intended for a one-year introductory course in abstract algebra with some topics of an
advanced level. Its design is such that the book can also be used for a one-semester course. The book
contains more material than normally would be taught in a one-year course. This should give the
teacher flexibility with respect to the selection of the content and the level at which the book is to be
used. We give a rigorous treatment of the fundamentals of abstract algebra with numerous examples
to illustrate the concepts. It usually takes students some time to become comfortable with the seeming
abstractness of modern algebra. Hence we begin at a leisurelypace paying great attention to the clarity
of our proofs. The only real prerequisite for the course is the appropriate mathematical maturity of the
students. Although the material found in calculus is independent of that of abstract algebra, a year of
calculusistypicallygivenasaprerequisite. Sincemanyoftheexamplesinalgebracomesfrommatrices,
we assume that the reader has some basic knowledge of matrix theory. The book should prepare the
student for higher level mathematics courses and computer science courses. We have many problems of
varying difficulty appearing after each section. We occasionally leave as an exercise the verification of
a certain point in a proof. However, we do not rely on exercises to introduce concepts which will be
needed later on in the text.
A distinguishing feature of the book is the Worked-Out Exercises which appear after every section.
These Worked-Out Exercises provide not only techniques of problem solving, but also supply additional
information to enhance the level of knowledge of the reader. The reader should study the Worked-
Out Exercises that are marked with along with the chapter. Those not marked with may be
♦ ♦
skippedduringthefirstreading. Sprinkledthroughoutthebookarecommentsdealingwiththehistorical
development of abstract algebra.
We welcome any comments concerning the text. The comments may be forwarded to the following
e-mail addresses: [email protected] or [email protected]
D.S. Malik
J. N. Mordeson
M.K. Sen
v
vi PREFACE
Contents
Preface v
List of Symbols ix
1 Sets, Relations, and Integers 3
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Binary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Introduction to Groups 35
2.1 Elementary Properties of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Permutation Groups 59
3.1 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Subgroups and Normal Subgroups 71
4.1 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Normal Subgroups and Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Homomorphisms and Isomorphisms of Groups 97
5.1 Homomorphisms of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Isomorphism and Correspondence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 The Groups D and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4 8
6 Direct Product of Groups 123
6.1 External and Internal Direct Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Introduction to Rings 129
7.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2 Some Important Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8 Subrings, Ideals, and Homomorphisms 145
8.1 Subrings and Subfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.2 Ideals and Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.3 Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9 Ring Embeddings 165
9.1 Embedding of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
vii
viii CONTENTS
10 Direct Sum of Rings 171
10.1 Complete Direct Sum and Direct Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11 Polynomial Rings 177
11.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12 Euclidean Domains 185
12.1 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.2 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
12.3 Prime and Irreducible Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
13 Unique Factorization Domains 199
13.1 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
13.2 Factorization of Polynomials over a UFD . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
13.3 Irreducibility of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
14 Maximal, Prime, and Primary Ideals 213
14.1 Maximal, Prime, and Primary Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15 Modules and Vector Spaces 221
15.1 Modules and Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
16 Field Extensions 229
16.1 Algebraic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
16.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
16.3 Algebraically Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
17 Multiplicity of Roots 247
17.1 Multiplicity of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
18 Finite Fields 257
18.1 Finite Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
References 261
List of Symbols
belongs to
∈
/ does not belong to
∈
subset
⊆
proper subset
⊂
contains
⊇
properly contains
⊃
∆ symmetric difference
A B set difference
\
(a,b) ordered pair
A complement of a set A
0
N set of positive integers
Z set of integers
Z# set of nonnegative integers
Q set of rational numbers
Q+ set of positive rational numbers
Q set of nonzero rational numbers
∗
R set of real numbers
R+ set of positive real numbers
R set of nonzero real numbers
∗
C set of complex numbers
C set of nonzero complex numbers
∗
(S) power set of the set S
P
union of sets
∪
intersection of sets
∩n number of combinations of n objects
i
taken i at a time
(cid:3) (cid:4)
n! n factorial
ab a divides b
|
a/b a does not divide b
|
gcd(a,b) greatest common divisor of a and b
lcm(a,b) least common multiple of a and b
n a a +a + +a
i=1 i 1 2 ··· n
a sum of all elements of S
(cid:83)a∈S congruence modulo n
n
≡(cid:83)
f :A B f is a function from a set Ainto a set B
→
f(x) image of xunder f
(f) domain of f
D
(f) image of f
I
g f composition of mappings g and f
f◦1 inverse of a mapping f
−
I I = 1,2,...,n
n n
{ }
f(A) f(A)= f(a) a A , Ais a set
{ | ∈ }
contained in the domain of the function f
f 1(B) f 1(B)= x X f(x) B ,
− −
{ ∈ | ∈ }
where f :X Y and B Y
→ ⊆
composition
◦
Π product
ix
x LIST OF SYMBOLS
M (R) set of all n n matrices over R
n
×
X number of elements in a set X
| |
G order of the group G
| |
(a) order of an element a
◦
Z set of integers modulo n
n
Z(G) center of the group G
G/H quotient group
aH,Ha left, right coset of a in H
aHa 1 aHa 1 = aha 1 h H
− − −
{ | ∈ }
[G:H] index of the subgroup H in G
K Klein 4-group
4
S symmetric group on n symbols
n
A alternating group on n symbols
n
D dihedral group of degree n
n
S the subgroup generated by S
h i
a the subgroup generated by a
h i
direct sum
⊕
N(H) Normalizer of H
C(a) centralizer of a
Ker f kernel of f
isomorphism
'
Aut(G) set of all automorphisms of the group G
Inn(G) set of all inner automorphisms of the group G
G stabilizer of a or isotropy group of a
a
Cl(a) conjugacy class of a
G commutator subgroup of the group G
0
G[n] set of all x G with nx=0,G is a group
∈
nG nG= nx x G
{ | ∈ }
C(R) center of the ring R
Q real quaternions
R
Z[√n] Z[√n]= a+b√n a,b Z ,n is a
{ | ∈ }
fixed positive integer
Z[i] Z[i]= a+bi a,b Z
{ | ∈ }
Z[i√n] Z[i√n]= a+bi√n a,b Z , nis a
{ | ∈ }
fixed positive integer
Q[√n] Q[√n]= a+b√n a,b Q , n is a
{ | ∈ }
fixed positive integer
Q[i] Q[i]= a+bi a,b Q
{ | ∈ }
Q[i√n] Q[i√n]= a+bi√n a,b Q , n is a
{ | ∈ }
fixed positive integer
a the left ideal generated by a
h il
a the right ideal generated by a
h ir
a the ideal generated by a
h i
R/I quotient ring
Q(R) quotient field of the ring R
R[x] polynomial ring in x
degf(x) degree of the polynomial f(x)
R[x ,x ,...,x ] polynomial ring in n indeterminates
1 2 n
√I radical of an ideal I
radR Jacobson radical of a ring R
F/K field extension
K(C) smallest subfield containing the subfield K
and the subset C of a field
[F :K] degree of the field F over the field K
GF(n) Galois field of n elements
G(F/K) Galois group of the field F over the field K
F fixed field of the group G
G
Φ (x) nth cyclotomic polynomial
n
P plane of the field F
F