1\flWILEY
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Introduction to
Abstract Algebra
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. I
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Introduction to
Abstract Algebra
Fourth Edition
W. Keith Nicholson
University of Calgary
Calgary, Alberta, Canada
@)WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
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Copyright 2012 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Nicholson, W. Keith.
Introduction to abstract algebra / W. Keith Nicholson. - 4th ed.
p. em.
Includes bibliographical references and index.
ISBN 978-1-118-13535-8 (cloth)
1. Algebra, Abstract. I. Title.
QA162.N53 2012
512' .02-dc23
2011031416
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
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Contents
PREFACE
ix
ACKNOWLEDGMENTS
xvii
NOTATION USED IN THE TEXT
xix
A SKETCH OF THE HISTORY OF ALGEBRA TO 1929
0
Preliminaries
0.1
0.2
0.3
0.4
1
2
1
Proofs I 1
Sets I 5
Mappings I 9
Equivalences I 17
Integers and Permutations
1.1
1.2
1.3
1.4
1.5
Induction I 24
Divisors and Prime Factorization
Integers Modulo n I 42
Permutations I 53
An Application to Cryptography
23
I
32
I
67
Groups
2.1
2.2
2.3
2.4
xxiii
69
Binary Operations I 70
Groups I 76
Subgroups I 86
Cyclic Groups and the Order of an Element
I
90
v
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vi
Contents
2.5
2.6
2. 7
2.8
2.9
2.10
2.11
3
Polynomials I 203
Factorization of Polynomials Over a Field I 214
Factor Rings of Polynomials Over a Field I 227
Partial Fractions I 236
Symmetric Polynomials I 239
Formal Construction of Polynomials I 248
251
Irreducibles and Unique Factorization
Principal Ideal Domains I 264
I
252
Fields
6.1
6.2
6.3
6.4
6.5
6.6
6. 7
7
202
Factorization in Integral Domains
5.1
5.2
6
Examples and Basic Properties I 160
Integral Domains and Fields I 171
Ideals and Factor Rings I 180
Homomorphisms I 189
Ordered Integral Domains I 199
Polynomials
4.1
4.2
4.3
4.4
4.5
4.6
5
159
Rings
3.1
3.2
3.3
3.4
3.5
4
Homomorphisms and Isomorphisms I 99
Cosets and Lagrange's Theorem I 108
Groups of Motions and Symmetries I 117
Normal Subgroups I 122
Factor Groups I 131
The Isomorphism Theorem I 137
An Application to Binary Linear Codes I 143
274
Vector Spaces I 275
Algebraic Extensions I 283
Splitting Fields I 291
Finite Fields I 298
Geometric Constructions I 304
The Fundamental Theorem of Algebra I 308
An Application to Cyclic and BCH Codes I 310
Modules over Principal Ideal Domains
7.1
7.2
Modules I 324
Modules Over a PID
I
335
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324
Contents
8
p-Groups and the Sylow Theorems
8.1
8.2
8.3
8.4
8.5
8.6
9
Products and Factors I 350
Cauchy's Theorem I 357
Group Actions I 364
The Sylow Theorems I 371
Semidirect Products I 379
An Application to Combinatorics
I
349
382
Series of Subgroups
9.1
9.2
9.3
388
The Jordan-Holder Theorem
Solvable Groups I 395
Nilpotent Groups 1 401
I
389
10 Galois Theory
10.1
10.2
10.3
10.4
412
Galois Groups and Separability I 413
The Main Theorem of Galois Theory I 422
Insolvability of Polynomials I 434
Cyclotomic Polynomials and Wedderburn's Theorem
11 Finiteness Conditions for Rings and Modules
11.1 Wedderburn's Theorem I 448
11.2 The Wedderburn-Artin Theorem
I
I
442
447
457
Appendices
Appendix
Appendix
Appendix
Appendix
vii
4 71
A Complex Numbers I 471
B Matrix Algebra I 478
C Zorn's Lemma I 486
D Proof of the Recursion Theorem
I
490
BIBliOGRAPHY
492
SELECTED ANSWERS
495
INDEX
523
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Preface
This book is a self-contained introduction to the basic structures of abstract algebra:
groups, rings, and fields. It is designed to be used in a two-semester course for
undergraduates or a one-semester course for seniors or graduates. The table of
contents is flexible (see the chapter summaries that follow), so the book is suitable for
a traditional course at various levels or for a more application-oriented treatment.
The book is written to be read by students with little outside help and so can be
used for self-study. In addition, it contains several optional sections on special topics
and applications.
Because many students will not have had much experience with abstract thinking, a
number of important concrete examples (number theory, integers modulo n, permutations) are introduced at the beginning and referred to throughout the book.
These examples are chosen for their importance and intrinsic interest and also because the student can do actual computations almost immediately even though the
examples are, in the student's view,. quite abstract. Thus, they provide a bridge
to the abstract theory and serve as prototype examples of the abstract structures
themselves. As an illustration, the student will encounter composition and inverses
of permutations before having to fit these notions into the general framework of
group theory.
The axiomatic development of these structures is also emphasized. Algebra provides
one of the best illustrations of the power of abstraction to strip concrete examples
of nonessential aspects and so to reveal similarities between ostensibly different
objects and to suggest that a theorem about one structure may have an analogue
for a different structure. Achieving this sort of facility with abstraction is one of the
goals of the book. This goes hand in hand with another goal: to teach the student
how to do proofs. The proofs of most theorems are at least as important for the
techniques as for the theorems themselves. Hence, whenever possible, techniques
are introduced in examples before giving them in the general case as a proof. This
partly explains the large number of examples (over 450) in the book.
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Preface
Of course, a generous supply of exercises is essential if this subject is to have a
lasting impact on students, and the book contains more than 1450 exercises (many
with separate parts). For the most part, computational exercises appear first, and
the exercises are given in ascending order of difficulty. Hints are given for the less
straightforward problems, and answers are provided to odd numbered (parts of)
computational exercises and to selected theoretical exercises. (A student solution
manual is available.) While exercises are vital to understanding this subject, they
are not used to develop results needed later in the text.
An increasing number of students of abstract algebra come from outside mathematics and, for many of them, the lure of pure abstraction is not as strong as for
mathematicians. Therefore, applications of the theory are included that make the
subject more meaningful and lively for these students (and for the mathematicians!).
These include cryptography, linear codes, cyclic and BCH codes, and combinatorics,
as well as "theoretical" applications within mathematics, such as the impossibility
of the classical geometric constructions. Moreover, the inclusion of short historical
notes and biographies should help the reader put the subject into perspective. In
the same spirit, some classical "gems" appear in optional sections (one example is
the elegant proof of the fundamental theorem of algebra in Section 6.6, using the
structure theorem for symmetric polynomials). In addition, the modern flavor of
the subject is conveyed by mentioning some unsolved problems and recent achievements, and by occasionally stating more advanced theorems that extend beyond
the results in the book.
Apart from that the material is quite standard. The aim is to reveal the basic
facts about groups, rings, and fields and give the student the working tools for
applications and further study. The level of exposition rises slowly throughout the
book and no prior knowledge of abstract algebra is required. Even linear algebra is
not needed. Except for a few well-marked instances, the aspects of linear algebra
that are needed are developed in the text. Calculus is completely unnecessary. Some
preliminary topics that are needed are covered in Chapter 0, with appendices on
complex numbers and matrix algebra (over a commutative ring).
Although the chapters are necessarily arranged in a linear order, this is by no
means true of the contents, and the student (as well as the instructor) should keep
the chapter dependency diagram in mind. A glance at that diagram shows that
while Chapters 1-4 are the core of the book, there is enough flexibility in the
remaining chapters to accommodate instructors who want to create a wide variety
of courses. The jump from Chapter 6 to Chapter 10 deserves mention. The student
has a choice at the end of Chapter 6: either change the subject and return to
group theory or continue with fields in Chapter 10 (solvable groups are adequately
reviewed in Section 10.3, so Chapter 9 is not necessary). The chapter summaries
that follow, and the chapter dependency diagram, can assist in the preparation of
a course syllabus.
Our introductory course at Calgary of 36 lectures touches Sections 0.3 and
0.4 lightly and then covers Chapters 1-4 except for Sections 1.5, 2.11, 3.5, and
4.4-4.6. The sequel course (also 36 lectures) covers Chapters 5, 6, 10, 7, 8, and 9,
omitting Sections 6.6, 6.7, 8.5, 8.6, and 10.4 and Chapter 11.
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Preface
xi
FEATURES
This book offers the following significant features:
• Self-contained treatment, so the book is suitable for self-study.
• Preliminary material for self-study or review available in Chapter 0 and in
Appendices A and B.
• Elementary number theory, integers modulo n, and permutations done first as
a bridge to abstraction.
• Over 450 worked examples to guide the student.
• Over 1450 exercises (many with parts), graded in difficulty, with selected
answers.
• Gradual increase in level throughout the text.
• Applications to number theory, combinatorics, geometry, cryptography, coding, and equations.
• Flexibility in syllabus construction and choice of optional topics (see chapter
dependency diagram).
• Historical notes and biographies.
• Several special topics (for example, symmetric polynomials, nilpotent groups,
and modules).
" Solution manual containing answers or solutions to all exercises.
• Student solution manual available with solutions to all odd numbered (parts
of) exercises.
CHANGES IN THE THIRD EDITION (2007)
The important concept of a module was introduced and used in Chapters 7 and 11.
• Chapter 7 on finitely generated abelian groups was completely rewritten, modules were introduced, direct sums were studied, and the rank of a free module
was defined (for commutative rings). Then the structure of finitely generated
modules over a PID was determined.
• Chapter 11 was upgraded from finite dimensional algebras to rings with the
descending chain condition. Wedderburn's characterization of simple artinian
rings and the Wedderburn-Artin theorem on semisimple rings were proved.
• A new section on semidirect products of groups was added.
" Appendices on Zorn's lemma and the recursion theorem were added.
• More solutions to theoretical exercises were included in the Selected Answers
section.
CHANGES IN THE FOURTH EDITION
The changes in the Third Edition primarily involved new concepts (modules, semidirect products, etc). However, the changes in the Fourth Edition are more
"microscopic" in nature, having more to do with clarity of exposition and making
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xii
Preface
the "flow" of arguments more natural and inevitable. Of course, minor editorial
changes are made through the book to correct typographical errors, improve the
exposition, and in some cases remove unnecessary material. Here are some more
specific changes.
• Because of the increasing importance of modules in the undergraduate curriculum, the new material on modules over a PID (Chapter 7) and the Wedderburn theorems (Chapter 11) introduced in the Third Edition was thoroughly
reviewed for clarity of exposition.
" More generally, in an effort to make the book more accessible to students, the
writing was carefully edited to ensure readability and clarity, the goal being
to make arguments flow naturally and, as much as possible, effortlessly. Of
course, this is in accord with the goal of making the book more suitable for
self-study.
" Appendix B is expanded to an exposition of matrix algebra over a commutative ring.
" Two notational changes are introduced. First, the symbol o(g) replaces lgl for
the order of an element g in a group, reducing confusion with the cardinality
lXI of a set. Second, polynomials f(x) are written simply as f.
• In Chapter 2, proofs of two early examples of "structure theorems" are given
to motivate the subject: A group of order 2p (p a prime) is cyclic or dihedral,
and an abelian group of order p 2 is Cp2 or Cp x Cp.
" More emphasis is placed on characteristic subgroups and on the product H K
of subgroups H and K.
" Wilson's theorem is included in §1.3 with later applications to number theory
and fields.
" In Chapter 5, it is shown that an integral domain is a UFD if and only if it
has the ACC on principal ideals and either (a) every irreducible is prime, or
(b) any two nonzero elements have a greatest common divisor. This shortens
the original proof (with (a) only) at the expense of a lemma of independent
interest.
• In Chapter 6, a simpler proof is given that any finite multiplicative subgroup
of a field is cyclic.
" The first section of Chapter 8 has been completely rewritten with several
results added.
" In Chapter 9, several new results on nilpotent groups have been included.
In particular, the Fitting subgroup of any finite group G is introduced, several properties are deduced, and its relationship to the Frattini subgroup is
explained.
" In Chapter 10, many arguments are rewritten and clarified, in particular the
lemma explaining the basic Galois connectionbetween the subgroups of the
Galois group of a field extension and the intermediate fields of the extension.
• In Chapter 11, a new elementary proof is given that R=Ln, where Lis a simple
left ideal of the simple ring R. This directly leads to Wedderburn's theorem,
and the proof does not involve the theory of semisimple modules.
• A student solution manual is now available giving detailed solutions to all odd
numbered (parts of) exercises.
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Preface
xiii
CHAPTER SUMMARIES
Chapter 0. Preliminaries. This chapter should be viewed as a primer on mathematics because it consists of materials essential to any mathematics major. The
treatment is self-contained. I personally ask students to read Sections 0.1 and 0.2,
and I touch briefly on the highlights of Sections 0.3 and 0.4. (Our students have
had complex numbers and one semester of linear algebra, so a review of Appendices
A and B is left to them.)
Chapter 1. Integers and Permutations. This chapter covers the fundamental
properties of the integers and the two prototype examples of rings and groups: the
integers modulo nand the permutation group Sn. These are presented naively and
allow the students to begin doing ring and group calculations in a concrete setting.
Chapter 2. Groups. Here, the basic facts of group theory are developed, including cyclic groups, Lagrange's theorem, normal subgroups, factor groups, homomorphisms, and the isomorphism theorem. The simplicity of the alternating groups An
is established for n 2: 5. An optional application to binary linear codes in included.
Chapter 3. Rings. The basic properties of rings are developed: integral domains,
characteristic, rings of quotients, ideals, factor rings, homomorphisms and the isomorphism theorem. Simple rings are studied, and it is shown that the ring of n x n
matrices over a division ring is simple.
Chapter 4. Polynomials. After the usual elementary facts are developed,
irreducible polynomials are discussed and the unique factorization of polynomials
over a field is proved. The factor rings of polynomials over a field are described
in detail, and some finite fields are constructed. In an optional section, symmetric
polynomials are discussed and the fundamental structure theorem is proved.
Chapter 5. Factorization in Integral Domains. Unique factorization domains
(UFDs) are characterized in terms of irreducibles, primes, and greatest common
divisors. The fact that being a UFD is inherited by polynomial rings is derived.
Principal ideal domains and euclidean domains are discussed. This chapter is selfcontained, and the material presented is not required elsewhere.
Chapter 6. Fields. After a minimal amount of vector space theory is developed,
splitting fields are constructed and used to completely describe finite fields. This
topic is a direct continuation of Section 4.3. In optional sections, the classical results on geometric constructions are derived, the fundamental theorem of algebra
is proved, and the theory of cyclic and BCH codes is developed.
Chapter 7. Modules over Principal Ideal Domains. Motivated by vector
spaces (Section 6.1) and abelian groups, the idea of a module over a ring is introduced. Free modules are discussed and the uniqueness of the rank is proved
for IBN rings. With abelian groups as the motivating example, the structure of
finitely generated modules over a principal ideal domain is determined, yielding the
fundamental theorem for finitely generated abelian groups.
Chapter 8. p-Groups and the Sylow Theorems. This chapter is a direct continuation of Section 2.10. After some preliminaries (including the correspondence
theorem), the class equation is developed and used to prove Cauchy's theorem
and to derive the basic properties of p-groups. Then group actions are introduced,
motivated by the class equation and an extended Cayley theorem, and used to prove
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Preface
the Sylow theorems. Semidirect products are presented. An optional application to
combinatorics is also included.
Chapter 9. Series of Subgroups. The chapter begins with composition series
and the Jordan-Holder theorem. Then solvable series are introduced, including the
derived series, and the basic properties of solvable groups are developed. Sections
9.1 and 9.2 depend only on the second and third isomorphism theorems and the
correspondence theorem in Section 8.1. Finally, in Section 9.3, central series are
discussed and nilpotent groups are characterized as direct products of p-groups,
and the Frattini and Fitting subgroups are introduced.
Chapter 10. Galois Theory. Galois groups of field extensions are defined, separable elements are introduced, and the main theorem of Galois theory is proved.
Then it is shown that polynomials of degree 5 or more are not solvable in radicals.
This requires only Chapter 6 (the reference to solvable groups in Section 10.3 is
adequately reviewed there). Finally, cyclotomic polynomials are discussed and used
(with the class equation) to prove Wedderburn's theorem that every finite division
ring is a field.
Chapter 11. Finiteness Conditions for Rings and Modules. The ascending and descending chain conditions on a module are introduced and the JordanHolder theorem is proved. Then endomorphism rings are used to prove Wedderburn's theorem that a simple, left artinian ring is a matrix ring over a division
ring. Next, semisimple modules are studied and the results are employed to prove
the Wedderburn-Artin theorem that a semisimple ring is a finite product of matrix
rings over division rings. In addition, it is shown that these semisimple rings are
characterized as the rings with every module projective and as the semiprime, left
artinian rings.
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Preface
Chapter Dependency Diagram
0 Preliminaries
I
1 Integers and
Permutations
2 Groups
3 Rings
5 Factorization
8 p-Groups and the
Sylow Theorem
6 Fields
-------------.
''
!
t
9 Series of
Subgroups
10 Galois Theory
A Dashed arrow indicates minor dependency.
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11 Finiteness
Conditions
xv
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Acknowledgments
I express my appreciation to the following people for their useful comments and
suggestions for the first edition of the book: F. Doyle Alexander, Stephen F. Austin
State University; Steve Benson, Saint Olaf College; Paul M. Cook II, Furman University; Ronald H. Dalla, Eastern Washington University; Robert Fakler, University
of Michigan-Dearborn; Robert M. Guralnick, University of Southern California;
Edward K. Hinson, University of New Hampshire; Ron Hirschorn, Queen's University; David L. Johnson, Lehigh University; William R. Nico, California State
University-Hayward; Kimmo I. Rosenthal, Union College; Erik Shreiner (deceased),
Western Michigan University; S. Thomeier, Memorial University; and Marie A.
Vitulli, University of Oregon.
I also want to thank all the readers who informed me about typographical and
other minor errors in the third edition. Particular thanks go to:
Carl Faith, Rutgers University, for giving the book a careful study and making
many very useful suggestions, too numerous to list here;
David French, Derbyshire, UK, for pointing out several typographical errors;
Michel Racine, Universite d'Ottawa, for pointing out a mistake in an exercise
deducing the commutativity of addition in a ring from the other axioms;
Yoji Yoshii, Universite d'Ottawa, for revealing two errors in the exercises
for Chapter 5;
Yiqiang Zhou, Memorial University of Newfoundland, for many helpful
suggestions and comments.
For the fourth edition, special thanks go to:
Jerome Lefebvre, University of Ottawa, for pointing out several typographical
errors;
xvii
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xviii
Acknowledgments
Edgar Goodaire and his students, Memorial University, for finding dozens of
typographical errors and making many useful suggestions;
Keith Conrad, University of Connecticut, for many useful comments on the
exposition;
Nazih Nahlus, American University of Beirut, for the proof that a finite
multiplicative group of a field is cyclic;
Matthew Greenberg, University of Calgary, for pointing out that Burnside's
lemma on Counting Orbits was due to Cauchy and Frobenius.
Milosz Kosmider, student, for correcting an error in Chapter 0;
Yannis Avrithis, National Technical University of Athens, for pointing out
dozens of typographical errors and making several suggestions.
It is a pleasure to thank Steve Quigley for his generous assistance throughout the
project. Thanks also go to the production staff at Wiley and particularly to Susanne
Steitz-Filler for keeping the project on schedule and responding so quickly to all
my questions. I also want to thank Joanne Canape for her vital assistance with the
computer aspects of the project.
Finally, I want to thank my wife, Kathleen, for her unfailing support. Without her
understanding and cooperation during the many hours that I was absorbed with
this project, this book would not exist.
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Notation used zn the Text
Symbol
Description
=}
implication
logical equivalence
set membership
set containment
proper set containment
set of natural numbers
set of integers
set of rational numbers
set of real numbers
set of complex numbers
positive elements in these sets
empty set
union of sets
intersection of sets
difference set
ordered pair
cartesian product of sets A and B
ordered n-tuple
2
3
mapping a from A to B
10
image of x under mapping a
image of mapping a
number of elements in set A
composite of mappings a and (3
identity mapping on set A
inverse of mapping a
equivalence relation
equivalence class of a
10
{:?
E
c
c
N
z
Q
lR
c
z+,Q+,JR+
0
u
n
A'-B
(a, b)
AxE
(a1, az, ... , an)
a:A-tB}
A~B
a(x)
im(a)
IAI
(3a
1A
a-1
=
[a]
First Used
5
5
5
5
5
5
6
6
6
6
7
7
7
7
8
8
12
12
12
13
14
17
17
xix
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Notation used in the Text
xx
Symbol
Description
A:
quotient set of equivalence
n factorial
binomial coefficient
d is a divisor of n
n!
(~)
din
gcd(~,n)
}
gcd(n1, ... ,nr)
lcm(~,n)
}
lcm( n1, ... , nr)
a= b (modn)
a
Zn
Sn
(:1 : :
E:
(k1 k2
·' · kr)
An
sgna
an
a-1
co
Un
M*
Sx
GLn(R)
Cn
K4
SLn(R)
PSLn(F)
Z(G)
(g)
o(g)
(X)
autG
innG
Ha, aH
IG:HI
Dn
H
Q
G/K
G'
kera
Bn
F(X,R)
Mn(R)
charR
First Used
=
19
26
26
33
greatest common divisor
33, 39
least common multiple
39
congruence modulo n
residue class of an integer a
integers modulo n
symmetric group of degree n
43
43
43
permutation a in Sn
54
identity permutation in Sn
cycle permutation in Sn
alternating group of degree n
sign of permutation a
nth power of a
inverse of a
circle group
group of nth roots of unity
group of units of monoid M
group of permutations of set X
general linear group over R
cyclic group of order n
Klein 4-group
special linear group over R
projective special linear group over F
center of group G
cyclic subgroup generated by g
order of group element g
subgroup generated by X
automorphism group of G
inner automorphism group of G
right, left cosets of subgroup H
index of subgroup H in G
dihedral group
H is a normal subgroup of G
quaternion group
factor group of G by K
derived (commutator) subgroup of G
kernel of a homomorphism a
set of binary n-tuples
ring of functions X --+ R
ring of n x n matrices over R
characteristic of a ring R
55
58
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54
62
66, 138
72
73
77
77
79
79
80
82
83
138
398
87
91
92
96
104
105
109
111
113
122
127
132
134
137
144
161
161
163
Notation used in the Text
Symbol
Description
Z(i)
T2(R)
Z(R)
ROP
lHI
ring of gaussian integers
upper triangular matrices over R
center of a ring R
oppositeving
quaternions
annihilator of element a
ring extension of a general ring R
ring of polynomials in x over R
degree of polynomial f
cyclotomic polynomials
associates in an integral domain
space spanned by v1, ... , Vn
dimension of vector space V
dimension of E over a subfield F
field generated over F by u1, ... , Un
field of algebraic numbers
formal derivative of f
Galois field of order pn
direct sum of modules
direct sum of n copies of module M
rank of free module M
torsion submodule of M
p-primary component of IV!
conjugacy class of a
normalizer of a subgroup X
core of a subgroup H
orbit of x generated by G
stabilizer of x
number of Sylow p-subgroups
semidirect product of K by H
composition length of G
higher derived subgroups for G
central series for group G
Frattini subgroup of G
Fitting subgroup of G
Galois group of E over F
automorphisms fixing subfield K
elements fixed by subgroup H
elementary symmetric polynomials
The Priifer group for a prime p
group of module homomorphisms
endomorphism ring of module M
homogeneous component
real, imaginary, part of z
conjugate, absolute value of z
notation for cos e+ i sine
ann( a)
Rl
R[x]
degf
ci>n(x)
arvb
span{v1 , ... , Vn}
dimV
[E:F]
F(u1, ... ,un)
A
f'
GF(pn)
N1 EB N2 EB · · · EB Nk
Mn
rankM
T(M)
M(p)
class a
N(X)
coreH
G·x
S(x)
np
KxeH
lengthG
Q(i)
Zi(G), ri(G)
ci>(G)
F(G)
gal(E: F)
K'
Ho
SR(Xl, X2, ... , Xn)
Zpoo
hom(M,N)
end (M)
H(K)
re z, imz
z, lzl
eiO
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First Used
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182
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253
277
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325, 329
325
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334, 336
337
358
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402, 403
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439
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A Sketch of the History of
Algebra to 1929
2500 BC Hieroglyphic numerals used in Egypt.
2400 BC Babylonians begin positional algebraic notation.
600 BC Pythagoreans discuss prime numbers.
250 Diophantus writes Arithmetica, using notation from which modern notation
evolved, and insists on exact solutions of equations in integers.
830 al-Khowarizmi writes Al-jabr, a textbook giving rules for solving linear and
quadratic equations.
1202 Leonardo of Pisa writes Liber abaci on arithmetic and algebraic equations.
1545 Tartaglia solves the cubic, and Cardano publishes the result in his Ars Magna.
Imaginary numbers are suggested.
1580 Viete uses vowels to represent unknown quantities, with consonants for
constants.
1629 Fermat becomes the founder of the modern theory of numbers.
1636 Fermat and Descartes invent analytic geometry, using algebra in geometry.
1749 Euler formulates the fundamental theorem of algebra.
1771 Lagrange solves the general cubic and quartic by considering permutations
of the roots.
1799 Gauss publishes his first proof of the fundamental theorem of algebra.
1801 Gauss publishes his Disquisitiones Arithmeticae.
1813 Ruffini claims that the general quintic cannot be solved by radicals.
1824 Abel proves that the general quintic cannot be solved by radicals.
1829 Galois introduces groups of substitutions.
1831 Galois sends his great memoir to the French Academie, but it is rejected.
1843 Hamilton discovers the quaternions.
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