Tài liệu Advanced Modern Algebra by Joseph J. Rotman Hardcover: 1040 pages Publisher: Prentice Hall; 1st pdf

Advanced Modern Algebra
by Joseph J. Rotman
Hardcover: 1040 pages
Publisher: Prentice Hall; 1st edition (2002); 2nd printing (2003)
Language: English
ISBN: 0130878685
Book Description
This book's organizing principle is the interplay between groups and rings,
where “rings” includes the ideas of modules. It contains basic definitions,
complete and clear theorems (the first with brief sketches of proofs), and
gives attention to the topics of algebraic geometry, computers, homology,
and representations. More than merely a succession of definition-theorem-proofs,

this text put results and ideas in context so that students can appreciate why
a certain topic is being studied, and where definitions originate. Chapter
topics include groups; commutative rings; modules; principal ideal domains;
algebras; cohomology and representations; and homological algebra. For
individuals interested in a self-study guide to learning advanced algebra and
its related topics.
Book Info
Contains basic definitions, complete and clear theorems, and gives attention
to the topics of algebraic geometry, computers, homology, and representations.
For individuals interested in a self-study guide to learning advanced algebra
and its related topics.
To my wife

Marganit
and our two wonderful kids,
Danny and Ella,
whom I love very much

Contents
Second Printing . viii
Preface ix
Etymology xii
Special Notation xiii
Chapter 1 Things Past 1
1.1. Some Number Theory 1

1.2.RootsofUnity 15
1.3. Some Set Theory . . 25
Chapter 2 Groups I 39
2.1. Introduction 39
2.2.Permutations 40
2.3. Groups . . . 51
2.4. Lagrange’s Theorem 62
2.5.Homomorphisms 73
2.6. Quotient Groups . . . 82
2.7.GroupActions 96
Chapter 3 Commutative Rings I 116
3.1. Introduction 116

3.2. First Properties . . . 116
3.3. Polynomials 126
3.4.GreatestCommonDivisors 131
3.5.Homomorphisms 143
3.6. Euclidean Rings . . . 151
3.7. Linear Algebra . . . 158
Vector Spaces . . . 159
Linear Transformations . . . 171
3.8.QuotientRingsandFiniteFields 182
v
vi
Contents

Chapter 4 Fields 198
4.1. Insolvability of the Quintic . . 198
Formulas and Solvability by Radicals 206
Translation into Group Theory 210
4.2. Fundamental Theorem of Galois Theory 218
Chapter 5 Groups II 249
5.1. Finite Abelian Groups 249
DirectSums 249
Basis Theorem . . 255
Fundamental Theorem . . . 262
5.2. The Sylow Theorems 269
5.3. The Jordan–H

¨
older Theorem . 278
5.4. Projective Unimodular Groups 289
5.5.Presentations 297
5.6. The Nielsen–Schreier Theorem 311
Chapter 6 Commutative Rings II 319
6.1. Prime Ideals and Maximal Ideals 319
6.2.UniqueFactorizationDomains 326
6.3.NoetherianRings 340
6.4.ApplicationsofZorn’sLemma 345
6.5.Varieties 376
6.6. Gr

¨
obner Bases 399
Generalized Division Algorithm . . . 400
Buchberger’s Algorithm . . 411
Chapter 7 Modules and Categories 423
7.1. Modules . . 423
7.2.Categories 442
7.3. Functors . . 461
7.4. Free Modules, Projectives, and Injectives 471
7.5. Grothendieck Groups 488
7.6.Limits 498
Chapter 8 Algebras 520

8.1. Noncommutative Rings 520
8.2.ChainConditions 533
8.3.SemisimpleRings 550
8.4. Tensor Products . . . 574
8.5.Characters 605
8.6. Theorems of Burnside and of Frobenius 634
Contents
vii
Chapter 9 Advanced Linear Algebra 646
9.1. Modules over PIDs . 646
9.2. Rational Canonical Forms . . . 666
9.3. Jordan Canonical Forms 675

9.4.SmithNormalForms 682
9.5. Bilinear Forms 694
9.6. Graded Algebras . . 714
9.7. Division Algebras . . 727
9.8. Exterior Algebra . . . 741
9.9. Determinants 756
9.10. Lie Algebras 772
Chapter 10 Homology 781
10.1. Introduction 781
10.2. Semidirect Products 784
10.3. General Extensions and Cohomology . 794
10.4. Homology Functors 813

10.5. Derived Functors . . 830
10.6.ExtandTor 852
10.7. Cohomology of Groups . . . 870
10.8. Crossed Products . . 887
10.9. Introduction to Spectral Sequences . . 893
Chapter 11 Commutative Rings III 898
11.1. Local and Global . . 898
11.2. Dedekind Rings . . 922
Integrality 923
Nullstellensatz Redux 931
Algebraic Integers . 938
Characterizations of Dedekind Rings . 948

Finitely Generated Modules over Dedekind Rings . . . 959
11.3. Global Dimension . 969
11.4. Regular Local Rings 985
Appendix The Axiom of Choice and Zorn’s Lemma A-1
Bibliography B-1
Index I-1
Second Printing
It is my good fortune that several readers of the first printing this book apprised me of
errata I had not noticed, often giving suggestions for improvement. I give special thanks to
Nick Loehr, Robin Chapman, and David Leep for their generous such help.
Prentice Hall has allowed me to correct every error found; this second printing is surely
better than the first one.

Joseph Rotman
May 2003
viii
Preface
Algebra is used by virtually all mathematicians, be they analysts, combinatorists, com-
puter scientists, geometers, logicians, number theorists, or topologists. Nowadays, ev-
eryone agrees that some knowledge of linear algebra, groups, and commutative rings is
necessary, and these topics are introduced in undergraduate courses. We continue their
study.
This book can be used as a text for the first year of graduate algebra, but it is much more
than that. It can also serve more advanced graduate students wishing to learn topics on
their own; while not reaching the frontiers, the book does provide a sense of the successes

and methods arising in an area. Finally, this is a reference containing many of the standard
theorems and definitions that users of algebra need to know. Thus, the book is not only an
appetizer, but a hearty meal as well.
When I was a student, Birkhoff and Mac Lane’s A Survey of Modern Algebra was the
text for my first algebra course, and van der Waerden’s Modern Algebra was the text for
my second course. Both are excellent books (I have called this book Advanced Modern
Algebra in homage to them), but times have changed since their first appearance: Birkhoff
and Mac Lane’s book first appeared in 1941, and van der Waerden’s book first appeared
in 1930. There are today major directions that either did not exist over 60 years ago, or
that were not then recognized to be so important. These new directions involve algebraic
geometry, computers, homology, and representations (A Survey of Modern Algebra has
been rewritten as Mac Lane–Birkhoff, Algebra, Macmillan, New York, 1967, and this

version introduces categorical methods; category theory emerged from algebraic topology,
but was then used by Grothendieck to revolutionize algebraic geometry).
Let me now address readers and instructors who use the book as a text for a beginning
graduate course. If I could assume that everyone had already read my book, A First Course
in Abstract Algebra, then the prerequisites for this book would be plain. But this is not a
realistic assumption; different undergraduate courses introducing abstract algebra abound,
as do texts for these courses. For many, linear algebra concentrates on matrices and vector
spaces over the real numbers, with an emphasis on computing solutions of linear systems
of equations; other courses may treat vector spaces over arbitrary fields, as well as Jordan
and rational canonical forms. Some courses discuss the Sylow theorems; some do not;
some courses classify finite fields; some do not.
To accommodate readers having different backgrounds, the first three chapters contain

ix
x
Preface
many familiar results, with many proofs merely sketched. The first chapter contains the
fundamental theorem of arithmetic, congruences, De Moivre’s theorem, roots of unity,
cyclotomic polynomials, and some standard notions of set theory, such as equivalence
relations and verification of the group axioms for symmetric groups. The next two chap-
ters contain both familiar and unfamiliar material. “New” results, that is, results rarely
taught in a first course, have complete proofs, while proofs of “old” results are usually
sketched. In more detail, Chapter 2 is an introduction to group theory, reviewing permuta-
tions, Lagrange’s theorem, quotient groups, the isomorphism theorems, and groups acting
on sets. Chapter 3 is an introduction to commutative rings, reviewing domains, fraction

fields, polynomial rings in one variable, quotient rings, isomorphism theorems, irreducible
polynomials, finite fields, and some linear algebra over arbitrary fields. Readers may use
“older” portions of these chapters to refresh their memory of this material (and also to
see my notational choices); on the other hand, these chapters can also serve as a guide for
learning what may have been omitted from an earlier course (complete proofs can be found
in A First Course in Abstract Algebra). This format gives more freedom to an instructor,
for there is a variety of choices for the starting point of a course of lectures, depending
on what best fits the backgrounds of the students in a class. I expect that most instruc-
tors would begin a course somewhere in the middle of Chapter 2 and, afterwards, would
continue from some point in the middle of Chapter 3. Finally, this format is convenient
for the author, because it allows me to refer back to these earlier results in the midst of a
discussion or a proof. Proofs in subsequent chapters are complete and are not sketched.

I have tried to write clear and complete proofs, omitting only those parts that are truly
routine; thus, it is not necessary for an instructor to expound every detail in lectures, for
students should be able to read the text.
Here is a more detailed account of the later chapters of this book.
Chapter 4 discusses fields, beginning with an introduction to Galois theory, the inter-
relationship between rings and groups. We prove the insolvability of the general polyno-
mial of degree 5, the fundamental theorem of Galois theory, and applications, such as a
proof of the fundamental theorem of algebra, and Galois’s theorem that a polynomial over
a field of characteristic 0 is solvable by radicals if and only if its Galois group is a solvable
group.
Chapter 5 covers finite abelian groups (basis theorem and fundamental theorem), the
Sylow theorems, Jordan–H

¨
older theorem, solvable groups, simplicity of the linear groups
PSL(2, k), free groups, presentations, and the Nielsen–Schreier theorem (subgroups of free
groups are free).
Chapter 6 introduces prime and maximal ideals in commutative rings; Gauss’s theorem
that R[x] is a UFD when R is a UFD; Hilbert’s basis theorem, applications of Zorn’s lemma
to commutative algebra (a proof of the equivalence of Zorn’s lemma and the axiom of
choice is in the appendix), inseparability, transcendence bases, L
¨
uroth’s theorem, affine va-
rieties, including a proof of the Nullstellensatz for uncountable algebraically closed fields
(the full Nullstellensatz, for varieties over arbitrary algebraically closed fields, is proved

in Chapter 11); primary decomposition; Gr
¨
obner bases. Chapters 5 and 6 overlap two
chapters of A First Course in Abstract Algebra, but these chapters are not covered in most
Preface
xi
undergraduate courses.
Chapter 7 introduces modules over commutative rings (essentially proving that all
R-modules and R-maps form an abelian category); categories and functors, including
products and coproducts, pullbacks and pushouts, Grothendieck groups, inverse and direct
limits, natural transformations; adjoint functors; free modules, projectives, and injectives.
Chapter 8 introduces noncommutative rings, proving Wedderburn’s theorem that finite

division rings are commutative, as well as the Wedderburn–Artin theorem classifying semi-
simple rings. Modules over noncommutative rings are discussed, along with tensor prod-
ucts, flat modules, and bilinear forms. We also introduce character theory, using it to prove
Burnside’s theorem that finite groups of order p
m
q
n
are solvable. We then introduce multi-
ply transitive groups and Frobenius groups, and we prove that Frobenius kernels are normal
subgroups of Frobenius groups.
Chapter 9 considers finitely generated modules over PIDs (generalizing earlier theorems
about finite abelian groups), and then goes on to apply these results to rational, Jordan, and

Smith canonical forms for matrices over a field (the Smith normal form enables one to
compute elementary divisors of a matrix). We also classify projective, injective, and flat
modules over PIDs. A discussion of graded k-algebras, for k a commutative ring, leads to
tensor algebras, central simple algebras and the Brauer group, exterior algebra (including
Grassmann algebras and the binomial theorem), determinants, differential forms, and an
introduction to Lie algebras.
Chapter 10 introduces homological methods, beginning with semidirect products and
the extension problem for groups. We then present Schreier’s solution of the extension
problem using factor sets, culminating in the Schur–Zassenhaus lemma. This is followed
by axioms characterizing Tor and Ext (existence of these functors is proved with derived
functors), some cohomology of groups, a bit of crossed product algebras, and an introduc-
tion to spectral sequences.

Chapter 11 returns to commutative rings, discussing localization, integral extensions,
the general Nullstellensatz (using Jacobson rings), Dedekind rings, homological dimen-
sions, the theorem of Serre characterizing regular local rings as those noetherian local
rings of finite global dimension, the theorem of Auslander and Buchsbaum that regular
local rings are UFDs.
Each generation should survey algebra to make it serve the present time.
It is a pleasure to thank the following mathematicians whose suggestions have greatly
improved my original manuscript: Ross Abraham, Michael Barr, Daniel Bump, Heng Huat
Chan, Ulrich Daepp, Boris A. Datskovsky, Keith Dennis, Vlastimil Dlab, Sankar Dutta,
David Eisenbud, E. Graham Evans, Jr., Daniel Flath, Jeremy J. Gray, Daniel Grayson,
Phillip Griffith, William Haboush, Robin Hartshorne, Craig Huneke, Gerald J. Janusz,
David Joyner, Carl Jockusch, David Leep, Marcin Mazur, Leon McCulloh, Emma Previato,

Eric Sommers, Stephen V. Ullom, Paul Vojta, William C. Waterhouse, and Richard Weiss.
Joseph Rotman
Etymology
The heading etymology in the index points the reader to derivations of certain mathematical
terms. For the origins of other mathematical terms, we refer the reader to my books Journey
into Mathematics and A First Course in Abstract Algebra, which contain etymologies of
the following terms.
Journey into Mathematics:
π, algebra, algorithm, arithmetic, completing the square, cosine, geometry, irrational
number, isoperimetric, mathematics, perimeter, polar decomposition, root, scalar, secant,
sine, tangent, trigonometry.
A First Course in Abstract Algebra:

affine, binomial, coefficient, coordinates, corollary, degree, factor, factorial, group,
induction, Latin square, lemma, matrix, modulo, orthogonal, polynomial, quasicyclic,
September, stochastic, theorem, translation.
xii
Special Notation
A algebraic numbers 353
A
n
alternating group on n letters 64
Ab category of abelian groups 443
Aff(1, k) one-dimensional affine group over a field k 125
Aut(G) automorphism group of a group G 78

Br(k), Br(E/k) Brauer group, relative Brauer group 737, 739
C complexnumbers 15
C
•
, (C
•
, d
•
) complex with differentiations d
n
: C
n

→ C
n−1
815
C
G
(x) centralizer of an element x inagroupG 101
D(R) global dimension of a commutative ring R 974
D
2n
dihedral group of order 2n 61
deg( f ) degree of a polynomial f (x) 126
Deg( f ) multidegree of a polynomial f (x

1
, ,x
n
) 402
det(A) determinant of a matrix A 757
dim
k
(V ) dimension of a vector space V over a field k 167
dim(R) Krull dimension of a commutative ring R 988
End
k
(M) endomorphism ring of a k-module M 527

F
q
finite field having q elements 193
Frac(R) fraction field of a domain R 123
Gal(E/ k) Galois group of a field extension E/k 200
GL(V ) automorphisms of a vector space V 172
GL(n, k) n × n nonsingular matrices, entries in a field k 179
H division ring of real quaternions 522
H
n
, H
n

homology, cohomology . . 818, 845
ht(p) height of prime ideal p 987
I
m
integers modulo m 65
I or I
n
identity matrix 173
√
I radical of an ideal I 383
Id(A) ideal of a subset A ⊆ k
n

382
im f image of a function f 27
irr(α, k) minimal polynomial of α over a field k 189
xiii
xiv
Special Notation
k algebraic closure of a field k 354
K
0
(R), K
0
(C) Grothendieck groups, direct sums . 491, 489

K

(C) Grothendieck group, short exact sequences 492
ker f kernel of a homomorphism f 75
lD(R) left global dimension of a ring R 974
Mat
n
(k) ring of all n × n matrices with entries in k 520
R
Mod category of left R-modules 443
Mod
R

category of right R-modules 526
N natural numbers ={integers n : n ≥ 0} 1
N
G
(H ) normalizer of a subgroup H inagroupG 101
O
E
ring of integers in an algebraic number field E 925
O(x) orbit of an element x 100
PSL(n, k) projective unimodular group = SL(n, k)/center 292
Q rational numbers
Q quaternion group of order 8 79

Q
n
generalized quaternion group of order 2
n
298
R real numbers
S
n
symmetric group on n letters 40
S
X
symmetric group on a set X 32

sgn(α) signum of a permutation α 48
SL(n, k) n × n matrices of determinant 1, entries in a field k 72
Spec(R) the set of all prime ideals in a commutative ring R 398
U(R) group of units in a ring R 122
UT(n, k) unitriangular n × n matrices over a field k 274
T I
3
I
4
, a nonabelian group of order 12 . . . 792
tG torsion subgroup of an abelian group G 267
tr(A) trace of a matrix A 610

V four-group 63
Var (I ) variety of an ideal I ⊆ k[x
1
, ,x
n
] 379
Z integers 4
Z
p
p-adicintegers 503
Z(G) center of a group G 77
Z(R) center of a ring R 523

[G : H ] index of a subgroup H ≤ G 69
[E : k] degree of a field extension E/k 187
S  T coproduct of objects in a category . . 447
S  T product of objects in a category 449
S ⊕ T external,internaldirectsum 250
K × Q direct product 90
K  Q semidirect product 790

A
i
directsum 451


A
i
direct product . . 451
Special Notation
xv
lim
←−
A
i
inverselimit 500
lim
−→

A
i
directlimit 505
G

commutator subgroup 284
G
x
stabilizer of an element x 100
G[m] {g ∈ G : mg = 0}, where G isanadditiveabeliangroup 267
mG {mg : g ∈ G}, where G isanadditiveabeliangroup 253
G

p
p-primary component of an abelian group G 256
k[x] polynomials 127
k(x) rational functions 129
k[[x]] formalpowerseries 130
kX polynomials in noncommuting variables 724
R
op
opposite ring 529
Ra or (a) principal ideal generated by a 146
R
×

nonzero elements in a ring R 125
H ≤ GHis a subgroup of a group G 62
H < GHis a proper subgroup of a group G 62
H ✁ GHis a normal subgroup of a group G 76
A ⊆ BAis a submodule (subring) of a module (ring)B 119
A  BAis a proper submodule (subring) of a module (ring)B 119
1
X
identity function on a set X
1
X
identity morphism on an object X 443

f : a → bf(a) = b 28
|X| number of elements in a set X
Y
[T ]
X
matrix of a linear transformation T relative to bases X and Y 173
χ
σ
character afforded by a representation σ 610
φ(n) Euler φ-function 21

n

r

binomial coefficient n!/r !(n −r)! 5
δ
ij
Kronecker delta δ
ij
=

1ifi = j;
0ifi = j.
a

1
, ,a
i
, ,a
n
list a
1
, ,a
n
with a
i
omitted

1
Things Past
This chapter reviews some familiar material of number theory, complex roots of unity, and
basic set theory, and so most proofs are merely sketched.
1.1 SOME NUMBER THEORY
Let us begin by discussing mathematical induction. Recall that the set of natural numbers
N is defined by
N ={integers n : n ≥ 0};
that is, N is the set of all nonnegative integers. Mathematical induction is a technique of
proof based on the following property of N:
Least Integer Axiom.

1
There is a smallest integer in every nonempty subset C of N.
Assuming the axiom, let us see that if m is any fixed integer, possibly negative, then
there is a smallest integer in every nonempty collection C of integers greater than or equal
to m.Ifm ≥ 0, this is the least integer axiom. If m < 0, then C ⊆{m, m +1, ,−1}∪N
and
C =

C ∩{m, m + 1, ,−1}

∪


C ∩N

.
If the finite set C ∩{m, m + 1, ,−1} = ∅, then it contains a smallest integer that is,
obviously, the smallest integer in C;ifC ∩{m, m +1, ,−1}=∅, then C is contained
in N, and the least integer axiom provides a smallest integer in C.
Definition. A natural number p is prime if p ≥ 2 and there is no factorization p = ab,
where a < p and b < p are natural numbers.
1
This property is usually called the well-ordering principle.
1
2

Things Past Ch. 1
Proposition 1.1. Every integer n ≥ 2 is either a prime or a product of primes.
Proof. Let C be the subset of N consisting of all those n ≥ 2 for which the proposition
is false; we must prove that C = ∅. If, on the contrary, C is nonempty, then it contains a
smallest integer, say, m. Since m ∈ C, it is not a prime, and so there are natural numbers
a and b with m = ab, a < m, and b < m. Neither a nor b lies in C, for each of them is
smaller than m, which is the smallest integer in C, and so each of them is either prime or a
product of primes. Therefore, m = ab is a product of (at least two) primes, contradicting
the proposition being false for m. •
There are two versions of induction.
Theorem 1.2 (Mathematical Induction). Let S(n) be a family of statements, one for
each integer n ≥ m, where m is some fixed integer. If

(i) S(m) is true, and
(ii) S(n) is true implies S(n + 1) is true,
then S(n) is true for all integers n ≥ m.
Proof. Let C be the set of all integers n ≥ m for which S(n) is false. If C is empty, we
are done. Otherwise, there is a smallest integer k in C.By(i),wehavek > m, and so there
is a statement S(k − 1).Butk − 1 < k implies k − 1 /∈ C,fork is the smallest integer in
C. Thus, S(k − 1) is true. But now (ii) says that S(k) = S([k − 1] + 1) is true, and this
contradicts k ∈ C [which says that S(k) is false]. •
Theorem 1.3 (Second Form of Induction). Let S(n) be a family of statements, one for
each integer n ≥ m, where m is some fixed integer. If
(i) S(m) is true, and
(ii) if S(k) is true for all k with m ≤ k < n, then S(n) is itself true,

then S(n) is true for all integers n ≥ m.
Sketch of Proof. The proof is similar to the proof of the first form. •
We now recall some elementary number theory.
Theorem 1.4 (Division Algorithm). Given integers a and b with a = 0, there exist
unique integers q and r with
b = qa +rand0 ≤ r < |a|.
Sketch of Proof. Consider all nonnegative integers of the form b − na, where n ∈ Z.
Define r to be the smallest nonnegative integer of the form b − na, and define q to be the
integer n occurring in the expression r = b −na.
If qa + r = q

a + r


, where 0 ≤ r

< |a|, then |(q − q

)a|=|r

− r |.Now0≤
|r

− r| < |a| and, if |q − q


| = 0, then |(q − q

)a|≥|a|. We conclude that both sides
are 0; that is, q = q

and r = r

. •
Sec. 1.1 Some Number Theory
3
Definition. If a and b are integers with a = 0, then the integers q and r occurring in the
division algorithm are called the quotient and the remainder after dividing b by a.

Warning! The division algorithm makes sense, in particular, when b is negative. A
careless person may assume that b and −b leave the same remainder after dividing by a,
and this is usually false. For example, let us divide 60 and −60 by 7.
60 = 7 ·8 + 4 and − 60 = 7 ·(−9) +3
Thus, the remainders after dividing 60 and −60 by 7 are different.
Corollary 1.5. There are infinitely many primes.
Proof. (Euclid) Suppose, on the contrary, that there are only finitely many primes. If
p
1
, p
2
, , p

k
is the complete list of all the primes, define M = ( p
1
···p
k
) + 1. By
Proposition 1.1, M is either a prime or a product of primes. But M is neither a prime
(M > p
i
for every i ) nor does it have any prime divisor p
i
, for dividing M by p

i
gives
remainder 1 and not 0. For example, dividing M by p
1
gives M = p
1
( p
2
···p
k
) + 1, so
that the quotient and remainder are q = p

2
···p
k
and r = 1; dividing M by p
2
gives M =
p
2
( p
1
p
3

···p
k
) + 1, so that q = p
1
p
3
···p
k
and r = 1; and so forth. This contradiction
proves that there cannot be only finitely many primes, and so there must be an infinite
number of them. •
Definition. If a and b are integers, then a is a divisor of b if there is an integer d with

b = ad. We also say that a divides b or that b is a multiple of a, and we denote this by
a | b.
There is going to be a shift in viewpoint. When we first learned long division, we
emphasized the quotient q; the remainder r was merely the fragment left over. Here, we
are interested in whether or not a given number b is a multiple of a number a, but we are
less interested in which multiple it may be. Hence, from now on, we will emphasize the
remainder. Thus, a | b if and only if b has remainder r = 0 after dividing by a.
Definition. A common divisor of integers a and b is an integer c with c | a and c | b.
The greatest common divisoror gcd of a and b, denoted by (a, b), is defined by
(a, b) =

0ifa = 0 = b

the largest common divisor of a and b otherwise.
Proposition 1.6. If p is a prime and b is any integer, then
( p, b) =

p if p | b
1 otherwise.
Sketch of Proof. A positive common divisor is, in particular, a divisor of the prime p, and
hence it is p or 1. •
4
Things Past Ch. 1
Theorem 1.7. If a and b are integers, then (a, b) = d is a linear combination of a and
b; that is, there are integers s and t with d = sa +tb.

Sketch of Proof. Let
I ={sa + tb : s, t ∈ Z}
(the set of all integers, positive and negative, is denoted by Z). If I ={0},letd be the
smallest positive integer in I ; as any element of I ,wehaved = sa +tb for some integers
s and t. We claim that I = (d), the set of all multiples of d. Clearly, (d) ⊆ I .Forthe
reverse inclusion, take c ∈ I . By the division algorithm, c = qd + r, where 0 ≤ r < d.
Now r = c − qd ∈ I , so that the minimality of d is contradicted if r = 0. Hence, d | c,
c ∈ (d), and I = (d). It follows that d is a common divisor of a and b, and it is the largest
such. •
Proposition 1.8. Let a and b be integers. A nonnegative common divisor d is their gcd if
and only if c | d for every common divisor c.
Sketch of Proof. If d is the gcd, then d = sa +tb. Hence, if c | a and c | b, then c divides

sa +tb = d. Conversely, if d is a common divisor with c | d for every common divisor c,
then c ≤ d for all c, and so d is the largest. •
Corollary 1.9. Let I be a subset of Z such that
(i) 0 ∈ I;
(ii) if a, b ∈ I , then a − b ∈ I;
(iii) if a ∈ I and q ∈ Z, then qa ∈ I.
Then there is a natural number d ∈ I with I consisting precisely of all the multiples of d.
Sketch of Proof. These are the only properties of the subset I in Theorem 1.7 that were
used in the proof. •
Theorem 1.10 (Euclid’s Lemma). If p is a prime and p | ab, then p | aorp| b. More
generally, if a prime p divides a product a
1

a
2
···a
n
, then it must divide at least one of the
factors a
i
.
Sketch of Proof. If p  a, then ( p, a) = 1 and 1 = sp + ta. Hence, b = spb + tab is a
multiple of p. The second statement is proved by induction on n ≥ 2. •
Definition. Call integers a and b relatively prime if their gcd (a, b) = 1.
Corollary 1.11. Let a, b, and c be integers. If c and a are relatively prime and if c | ab,

then c | b.
Sketch of Proof. Since 1 = sc + ta,wehaveb = scb + tab. •
Sec. 1.1 Some Number Theory
5
Proposition 1.12. If p is a prime, then p



p
j

for 0 < j < p.

Sketch of Proof. By definition, the binomial coefficient

p
j

= p!/j!( p − j )!, so that
p! = j!( p − j )!

p
j

.

By Euclid’s lemma, p  j!( p − j)! implies p |

p
j

. •
If integers a and b are not both 0, Theorem 1.7 identifies (a, b) as the smallest positive
linear combination of a and b. Usually, this is not helpful in actually finding the gcd, but
the next elementary result is an exception.
Proposition 1.13.
(i) If a and b are integers, then a and b are relatively prime if and only if there are
integers s and t with 1 = sa +tb.

(ii) If d = (a, b), where a and b are not both 0, then (a/d, b/d) = 1.
Proof. (i) Necessity is Theorem 1.7. For sufficiency, note that 1 being the smallest posi-
tive integer gives, in this case, 1 being the smallest positive linear combination of a and b,
and hence (a, b) = 1. Alternatively, if c is a common divisor of a and b, then c | sa +tb;
hence, c | 1, and so c =±1.
(ii) Note that d = 0 and a/d and b/d are integers, for d is a common divisor. The equation
d = sa +tb now gives 1 = s(a/d) +t (b/d). By part (i), (a/d, b/d) = 1. •
The next result offers a practical method for finding the gcd of two integers as well as
for expressing it as a linear combination.
Theorem 1.14 (Euclidean Algorithm). Let a and b be positive integers. There is an
algorithm that finds the gcd,d = (a, b), and there is an algorithm that finds a pair of
integers s and t with d = sa +tb.

Remark. More details can be found in Theorem 3.40, where this result is proved for
polynomials.
To see how the Greeks discovered this result, see the discussion of antanairesis in
Rotman, A First Course in Abstract Algebra, page 49.

Sketch of Proof. This algorithm iterates the division algorithm, as follows. Begin with
b = qa +r , where 0 ≤ r < a. The second step is a = q

r +r

, where 0 ≤ r


< r; the next
step is r = q

r

+r

, where 0 ≤ r

< r

, and so forth. This iteration stops eventually, and

the last remainder is the gcd. Working upward from the last equation, we can write the gcd
as a linear combination of a and b. •
6
Things Past Ch. 1
Proposition 1.15. If b ≥ 2 is an integer, then every positive integer m has an expression
in base b: There are integers d
i
with 0 ≤ d
i
< b such that
m = d
k

b
k
+ d
k−1
b
k−1
+···+d
0
;
moreover, this expression is unique if d
k
= 0.

Sketch of Proof. By the least integer axiom, there is an integer k ≥ 0 with b
k
≤ m <
b
k+1
, and the division algorithm gives m = d
k
b
k
+r, where 0 ≤ r < b
k
. The existence of

b-adic digits follows by induction on m ≥ 1. Uniqueness can also be proved by induction
on m, but one must take care to treat all possible cases that may arise. •
The numbers d
k
, d
k−1
, ,d
0
are called the b-adic digits of m.
Theorem 1.16 (Fundamental Theorem of Arithmetic). Assume that an integer a ≥ 2
has factorizations
a = p

1
···p
m
and a = q
1
···q
n
,
where the p’s and q’s are primes. Then n = m and the q’s may be reindexed so that
q
i
= p

i
for all i . Hence, there are unique distinct primes p
i
and unique integers e
i
> 0
with
a = p
e
1
1
···p

e
n
n
.
Proof. We prove the theorem by induction on , the larger of m and n.
If  = 1, then the given equation is a = p
1
= q
1
, and the result is obvious. For the
inductive step, note that the equation gives p
m

| q
1
···q
n
. By Euclid’s lemma, there is
some i with p
m
| q
i
.Butq
i
, being a prime, has no positive divisors other than 1 and

itself, so that q
i
= p
m
. Reindexing, we may assume that q
n
= p
m
. Canceling, we have
p
1
···p

m−1
= q
1
···q
n−1
. By the inductive hypothesis, n − 1 = m − 1 and the q’s may
be reindexed so that q
i
= p
i
for all i. •
Definition. A common multiple of integers a and b is an integer c with a | c and b | c.

The least common multiple or lcm of a and b, denoted by [a, b], is defined by
[a, b] =

0ifa = 0 = b
the smallest positive common multiple of a and b otherwise.
Proposition 1.17. Let a = p
e
1
1
···p
e
n

n
and let b = p
f
1
1
···p
f
n
n
, where e
i
≥ 0 and f

i
≥ 0
for all i; define
m
i
= min{e
i
, f
i
} and M
i
= max{e

i
, f
i
}.
Then the gcd and the lcm of a and b are given by
(a, b) = p
m
1
1
···p
m
n

n
and [a, b] = p
M
1
1
···p
M
n
n
.
Sketch of Proof. Use the fact that p
e

1
1
···p
e
n
n
| p
f
1
1
···p
f

n
n
if and only if e
i
≤ f
i
for
all i. •
Sec. 1.1 Some Number Theory
7
Definition. Let m ≥ 0 be fixed. Then integers a and b are congruent modulo m, denoted
by

a ≡ b mod m,
if m | (a − b).
Proposition 1.18. If m ≥ 0 is a fixed integer, then for all integers a, b, c,
(i) a ≡ a mod m;
(ii) if a ≡ b mod m, then b ≡ a mod m;
(iii) if a ≡ b mod m and b ≡ c mod m, then a ≡ c mod m.
Remark. (i) says that congruence is reflexive, (ii) says it is symmetric, and (iii) says it is
transitive.

Sketch of Proof. All the items follow easily from the definition of congruence. •
Proposition 1.19. Let m ≥ 0 be a fixed integer.
(i) If a = qm +r, then a ≡ r mod m.

(ii) If 0 ≤ r

< r < m, then r ≡r

mod m; that is, r and r

are not congruent mod m.
(iii) a ≡ b mod m if and only if a and b leave the same remainder after dividing by m.
(iv) If m ≥ 2, each integer a is congruent mod m to exactly one of 0, 1, ,m − 1.
Sketch of Proof. Items (i) and (iii) are routine; item (ii) follows after noting that
0 < r −r


< m, and item (iv) follows from (i) and (ii). •
The next result shows that congruence is compatible with addition and multiplication.
Proposition 1.20. Let m ≥ 0 be a fixed integer.
(i) If a ≡ a

mod m and b ≡ b

mod m, then
a + b ≡ a

+ b


mod m.
(ii) If a ≡ a

mod m and b ≡ b

mod m, then
ab ≡ a

b

mod m.
(iii) If a ≡ b mod m, then a

n
≡ b
n
mod m for all n ≥ 1.
Sketch of Proof. All the items are routine. •
8
Things Past Ch. 1
Earlier we divided 60 and −60 by 7, getting remainders 4 in the first case and 3 in the
second. It is no accident that 4 +3 = 7. If a is an integer and m ≥ 0, let a ≡ r mod m and
−a ≡ r

mod m. It follows from the proposition that

0 =−a + a ≡ r

+r mod m.
The next example shows how one can use congruences. In each case, the key idea is to
solve a problem by replacing numbers by their remainders.
Example 1.21.
(i) Prove that if a is in Z, then a
2
≡ 0, 1, or 4 mod 8.
If a is an integer, then a ≡ r mod 8, where 0 ≤ r ≤ 7; moreover, by Proposi-
tion 1.20(iii), a
2

≡ r
2
mod 8, and so it suffices to look at the squares of the remainders.
r 0 1 2 3 4 5 6 7
r
2
0 1 4 9 16 25 36 49
r
2
mod 8 0 1 4 1 0 1 4 1
Table 1.1. Squares mod 8
We see in Table 1.1 that only 0, 1, or 4 can be a remainder after dividing a perfect square

by 8.
(ii) Prove that n = 1003456789 is not a perfect square.
Since 1000 = 8 · 125, we have 1000 ≡ 0 mod 8, and so
n = 1003456789 = 1003456 ·1000 + 789 ≡ 789 mod 8.
Dividing 789 by 8 leaves remainder 5; that is, n ≡ 5 mod 8. Were n a perfect square, then
n ≡ 0, 1, or 4 mod 8.
(iii) If m and n are positive integers, are there any perfect squares of the form 3
m
+3
n
+1?
Again, let us look at remainders mod 8. Now 3

2
= 9 ≡ 1 mod 8, and so we can evaluate
3
m
mod 8 as follows: If m = 2k, then 3
m
= 3
2k
= 9
k
≡ 1 mod 8; if m = 2k + 1, then
3

m
= 3
2k+1
= 9
k
· 3 ≡ 3 mod 8. Thus,
3
m
≡

1mod8 ifm is even;
3mod8 ifm is odd.

Replacing numbers by their remainders after dividing by 8, we have the following possi-
bilities for the remainder of 3
m
+ 3
n
+ 1, depending on the parities of m and n:
3 +1 +1 ≡ 5mod8
3 +3 +1 ≡ 7mod8
1 +1 +1 ≡ 3mod8
1 +3 +1 ≡ 5mod8.
Sec. 1.1 Some Number Theory
9

In no case is the remainder 0, 1, or 4, and so no number of the form 3
m
+ 3
n
+ 1 can be a
perfect square, by part (i).

Proposition 1.22. A positive integer a is divisible by 3(or by 9) if and only if the sum of
its (decimal) digits is divisible by 3(or by 9).
Sketch of Proof. Observe that 10
n
≡ 1 mod 3 (and also that 10

n
≡ 1 mod 9). •
Proposition 1.23. If p is a prime and a and b are integers, then
(a + b)
p
≡ a
p
+ b
p
mod p.
Sketch of Proof. Use the binomial theorem and Proposition 1.12. •
Theorem 1.24 (Fer m at). If p is a prime, then

a
p
≡ a mod p
for every a in Z. More generally, for every integer k ≥ 1,
a
p
k
≡ a mod p.
Sketch of Proof. If a ≥ 0, use induction on a; the inductive step uses Proposition 1.23.
The second statement follows by induction on k ≥ 1. •
Corollary 1.25. Let p be a prime and let n be a positive integer. If m ≥ 0 and if  is the
sum of the p-adic digits of m, then

n
m
≡ n

mod p.
Sketch of Proof. Write m in base p, and use Fermat’s theorem. •
We compute the remainder after dividing 10
100
by 7. First, 10
100
≡ 3
100

mod 7.
Second, since 100 = 2 · 7
2
+ 2, the corollary gives 3
100
≡ 3
4
= 81 mod 7. Since
81 = 11 ×7 + 4, we conclude that the remainder is 4.
Theorem 1.26. If (a, m) = 1, then, for every integer b, the congruence
ax ≡ b mod m
can be solved for x; in fact, x = sb, where sa ≡ 1modm is one solution. Moreover, any

two solutions are congruent mod m.
Sketch of Proof. If 1 = sa + tm, then b = sab + tmb. Hence, b ≡ a(sb) mod m. If,
also, b ≡ ax mod m, then 0 ≡ a(x −sb) mod m, so that m | a(x −sb). Since (m, a) = 1,
we have m | (x − sb); hence, x ≡ sb mod m, by Corollary 1.11. •