Higher algebra for the undergraduate

171 Pages·1949·19.109 MB·English

by  Marie Weiss

E R
H I G H
ALGEBRA
for the Undergraduate
MARIE J. WEISS
Professor of Mathematics
Newcomb College
Tulane University
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NEW YORK · JOHN WILEY & SONS, INC.
LONDON· CHAPMAN & HALL, LIMITED
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Copyright, 1949
by
JOHN WILEY & SONS, INC.
All Rights &served
Thia book or an11 part thereof mu,t not
be reproduced in an11 form witlund
the wriUen permiuion of the publiaher.
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Printed in the United St.aw, of America
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PREFACE
This textbook is intended for a six semester-hour course in
higher algebra for the undergraduate who has had two years of
college mathematics including calculus. Although practically no
knowledge of calculus is needed, the mathematical maturity de
veloped by its study is necessary. It is my belief that it is both
mathematically necessary and culturally desirable to introduce
the undergraduate at an early stage to some of the simpler algebraic
concepts that are as much a part of mathematics today as are the
~
elementary concepts of calculus. Consequently, such topics as
' .
' groups, rings, fields, and matrices are given an equal place with the
·-· theory of equations. Naturally, part of the traditional material
.,. in the theory of equations, such as the approximations to real roots,
l: has been omitted. I have found that the subject matter selected
presents no more difficulty to the average student than does the
traditional course in the theory of equations. Only the elements
of each concept and theory have been given. The book is intended
to serve merely as an introduction to algebraic concepts so that
the undergraduate may have some idea of the kind of concepts
used in that part of mathematics usually called algebra.
The presentation is some,vhat the same as in an elementary
course in calculus. Examples and exercises are given throughout.
The student is expected to work exercises for each class.meeting.
The number of ideas introduced at one time has been kept to a
minimum. Consequently, in the discussion of matrices, for
example, the concept of a vector space has not been introduced
~ an important omission. In general, the presentation is intended
for the average undergraduate who finds the more advanced texts
in higher algebra too difficult to read.
The book begins with a discussion of the number system which
emphasizes those properties that are to be used for illustrative
material in the elementary theory of groups and rings and which
orients the student to the algebraic point of view. The real num
bers are almost completely neglected. The elementary properties
y
4
vi PREFACE
of groups, rings, and fields are then developed. In order that the
student may have simple examples and use a noncommutative
operation, permutation groups are introduced in the chapter on
groups. The course proceeds with the elementary properties of
polynomials over a field, emphasizing their analogy with the prop
erties of integers. The elementary theory of matrices over a
field including the applicatjon to the solution of simultaneous
linear equations over a field is developed before determinants are
introduced as values associated with square matrices. The theory
of determinants and its connection with matrices then follow. The
book closes with a chapter that introduces the student to factor
groups, residue class rings, and the homomorphism theory of
groups and rings.
In an address given before the Mathematical Association of
America in 1939 ("Algebra for the Undergraduate," American
Mathematical Monthly, Vol. 46, pp. 635-642, 1939), I first outlined
the material for such a course, which I then had ·given for several
years to juniors. My indebtedness at that time to the standard
textbooks in higher algebra, particularly B. L. van der \Vaerden's
Moderne Algebra, A. A. Albert's Modern }Jigher Algebra, and H.
Hasse's Hohere Algebra, was evident. The more recent excellent
expositions given in G. Birkhoff and S. ~1acLane's A Survey of
Modern Algebra and in C. C. MacDuffee's two books, An Intro
duction lo Abstract Algebra and Vectors and Matrices, have influenced
my selection of proofs. Mention should also be made of the stand
ard textbooks in number theory, group theory, and theory of equa
tions. My students over a period of years have by their criticisms
and difficulties helped me in making choices of methods of exposi
tion. I am also indebted to Professor M. Gweneth Humphreys,
who read the first draft of the manuscript and made many helpful
suggestions.
MiltE J. WEt.SS
NEW ORLEANS, LoUISIANA
Odober 10, 1948
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CONTENTS
CHAPTER PAGE
1. The Integers 1
The positive integers, 1; Further properties, 3;
Finite induction, 4; Summary, 5; The integers, 6;
The number zero, 8; The positive integers as a sub
set of the integers, 9; The negative integers, 10;
Inequalities, 11; Division of integers, 12; Greatest
common divisor, 13; Prime factors, 16; Congruences,
17; The linear congruence, 19; Residue classes, 21;
Positional notation for integers, 22
2. The Rational, Real, and Complex Numbers 25
The rational numbers, 25; The integers as a subset of
the rational numbers, 27; The real numbers, 27; The
complex numbers, 31; The real numbers as a subset of
+
the complex numbers, 32; The notation a bi, 32;
Geometric representation of complex numbers, 33;
De Moivre's theorem, 34; Then nth roots of a com
plex number, 35; Primitive nth roots of unity, 37
3. Elementary Theory of Groups 39
Definition, 39; Elementary properties, 41; Permuta
tions, 42; Even and odd permutat.ions, 45; Isomor
phism, 47; Cyclic groups, 48; Subgroups, 51; Cosets
and subgroups, 53; Cayley's theorem, 56
4. Rings, Integral Domains, and Fields 58
Rings, 58; Integral domains and fields, 60; Quotients
in a field, 61; Quotient field, 62; Polynomials over
an integral domain, 64; Characteristic of an integral
domain, 65; Division in an integral domain, 67
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6
viii CONTENTS
CHAPTER PAGE
5. Polynomials over a Field 70
Division algorithm, 70; · Synthetic division, 72;
Greatest common divisor, 73; Factorization theorems,
77; Zeros of a polynomial, 79; Relation between the
zeros and coefficients of a polynomial, 82; Derivative
of a polynomial, 84; Multiple factors, 85; Taylor's
theorem for the polynomial, 87
6. Matrices over a Field 90
Matrix notation, 90; Addition and multiplication, 91;
Matrix multiplication a.nd systems of linear equations,
94; Special matrices, 96; Partitioning of.matrices, 98;
Row equivalence, 99; Nonsingular matrices, 104;
Column equivalence, 107; Equivalence of matrices,
108; Linear independence and dependence over a.
field, 109; Rank of a matrix, 111; Simultaneous linear
equations over a field, 114; Homogeneous linear equa
tions, 119; Linearly independent solutions of systems
of linear equations, 120
7. Determinants and Matrices 123
Definition, 123; Cofactors, 124; Further properties,
126; Laplace's expansion of a determinant, 131; Prod
ucts of determinants, 133; Adjoint and inverse of a.
matrix, 135; Cramer's rule, 137; Determinant rank
of a matrix, 138; Polynomials with matrix coefficients,
139; Similar matrices over a field, 142
8. Groups, Rings, and Ideals 145
Normal subgroups and factor groups, 145; Conjugates,
147; Automorphisms of a group, 149; Homomorphisms
of groups, 151; Ideals in commutative rings, 154;
Residue class rings, 156; Homomorphisms of rings,
158
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7
The Integers
1 • The positive integers
The mathematical symbols first encountered by everyone are those
of the positive integers: 1, 2, 3, , · · . These are often called the
natural numbers. Their properties are familiar to all, and we
shall list them systematically. It is not our purpose to develop
these properties from a minimum number of hypotheses and un
defined terms but rather to list those laws and properties that have
long been familiar to the student and to use them as a characterizing
definition of the positive integers.
The familiar operations on the positive integers are those of
addition and multiplication; that is, for every pair of positive
+
integers a, b we know what is meant by the sum a b and the
product ab and that the sum and product are again positive
integers. The fact that the sum and product of any two positive
integers are again positive integers is often expressed by saying
that the set of positive integers is closed under addition and multi
plication. As is well known, the positive integers a, b, c, · · · obey
the following laws governing these operations:
the commutative law
for addition a+b=b+a,
for multiplication ab= ba;
the associative law
for addition a+ (b + c) = (a+ b) + c, I
for multiplication a(bc) = (ab)c;
+
the distributive law a(b c) = ab+ ac.
Note the meaning of the parentheses in the associative and
1
8
2 THE INTEGERS
distributive laws, and note that, if the commutative law for multi
plication did not hold, it wo·uld be necessary to have a second or
+ = +
right-hand distributive law, (b c)a ba ca. As we see,
. however, this result could be obtained from the distributive law by
applying the commutative law for multiplication to both sides of
the equality. We shall study systems in which some of these laws
do not hold, and therefore it is necessary to have a clear under
standing of their meaning. Let us illustrate them with a fe,v
examples:
The equality
=
7(3 · 6) 6(3 · 7)
holds, for, by first applying the associative law and then applying
the commutative law for multiplication twice, we have
7 (3 · 6) = (7 · 3 )6 = 6(7 · 3) = 6(3 · 7).
Again, if a, b, c are positive integers, the equality
+
= +
a(b c) ca ab
holds, for, applying in succession the distributive law, the com
mutative law for addition, and the commutative law for multi
plication, we have
+ + +
a(b c) = ab ac = ac ab = ca+ ab.
An illustration of a system in which some of these laws do not
hold can be given by arbitrarily defining an addition and a multi
plication for positive integers as follows: Denote the new addition
by (±) and the new multiplication by O . Let a (±) b = 2a and
a O b = 2ab, where 2a and 2ab denote the results of ordinary
multiplication. Then
b (±) a = 2b, b O a = 2ba, a (±) (b (±) c) = a (±) 2b = 2a,
= = = =
(a© b) © c 2a © c 4a, a O (b O c) a O 2bc 4abc,
(a Ob) 0 c = 2ab O c = 4abc, a O (b (±) c) = a O 2b = 4ab,
= =
(a Ob)(±) (a O c) 2ab (±) 2ac 4ab.
Note that the commutative and associative laws fail for addition
but hold for multiplication. Are there two distributive laws in
this system?
9
FURTHER PROPERTIES 3
Exercises
1. Reduce the left-hand side of the following equalities to the right-hand
side by using successively one associative, commutative, or distributive law:
+ + = + + + = +
a) (3 5) 6 3 (5 6). b) 1 5 5 1.
= =
c) 2(3 · 5) (2 · 3)5. d) 2(3 · 5) 5(2 · 3).
+ = +
e) 6(8 4) 4 · 6 6 • 8. f) 6 (8 · 4) - ( 4 · 6 )8.
g) 3(7 + 5) = 5 · 3 + 7 · 3. h) 5(6 + 3) ~ 3 · 5 + 5 · 6.
= + = +
i) 6 (5 · 3) (3 · 6 )5. j) 4 · 6 7 · 4 4(7 6).
k) a(b + (c + d)] = (ab+ ac) + ad. I) a(b(cd)] = (bc)(ad).
=
m) a(b(cd)] (ab )(cd).
+ = + +
n) (ad+ ca) ag a[(g c) d].
2. Determine whether the operations @ and O for positive integers x, y
defined as follows obey the commutative, associative, and distributive laws:
= + O =
a) x @ y x 2y, x y 2xy.
+
b) X (:t) Y = X y2, XO y ~ xy2.
c) x @ Y = x2 + yz, x O Y = x2y2.
2 • Further properties
Some further properties of the positive integers will be listed.
Note that the positive integer 1 is the only positive integer such
that 1 · a = a, for every positive integer a. \Ve say that 1 is an
identity for multiplication. Again the following cancellation laws
for addition and multiplication hold:
+ +
1) if a x = b x, then a = b ;
= =
2) if ax bx, then a b.
Moreover, for any two positive integers a and b, either a = b, or
+
there exists a positive integer x such that a x = b, or there
+
exists a positive integer y such that a = b y.
From these alternative relations between two positive integers,
we can define inequalities. If a + x = b, we write a < b (read
>
a less than b) and b a (read b greater than a). Hence, for any
two positive integers, we have the following mutually exclusive
= < >
alternatives-either a b, or a b, or a b-and we have
established an order relation between any two positive integers.
From the above definition we may prove the familiar properties of
10
.4 THE INTEGERS
inequalities for positive integers:
1) if a < b and b < c, then a < c;
+ +
< <
2) if a b, then a c b c;
3) if a < b, then ac < be.
Their proof is left to the student.
3 · Finite induction
We come now to the last important property of the positive
integers that will be discussed. This property will enable us to
make proofs by the method known as finite induction or mathe
matical induction.
Postulate of finite induction. A set S of positive integers with
the following two properties contains all the positive integers:
. .
a) the set S contains the positive integer 1;
b) if the set S contains the positive integer k, it contains the
+
positive integer k 1.
This postulate is used to prove either true or false certain propo
sitions that involve all positive integers. The proof is said to be
made by finite induction.
First method of '{YT"oof by finite induction. Let P(n) be a propo
sition that is defined for every positive integer n. If P (1) is true,
+
and if P(k 1) is true whenever P(k) is true, then P(n) is true
for all positive integers n.
The proof is immediate by the postulate of finite induction.
For consider the set S of positive integers for which the proposition
P(n) is true. By hypothesis it contains the positive integer 1
+
and the positive integer k 1 whenever it contains the positive
integer k. Hence the set S contains all the positive integers.
Example. Let the power an, where n is a positive integer, be
defined as follows : a1 = a, ak+I = ak · a. Prove that (ab)" =
a"b".
If n = 1, we have (ab )1 = ab = a1b1 by the definition. Assume
that this law of exponents holds for n = k: (ab )k = akbk. Then
(ab/ (ab) = (ab )"'+1 by definition, and (ab)"' (ab) = (a"'b"') (ab) by
assumption. Applying the associative and commutative laws to
the right-hand side of the last equation and using the definition,
we have (ab)"'(ab) = (aka)(bkb) = ak+1b"+1 which was to be
,
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