First Course in Algebra and Number Theory

551 Pages·1971·19.743 MB·English

by  Edwin Weiss (Auth.)

FIRST COURSE IN
ALGEBRA AND
NUMBER THEORY
Edwin Weiss
Boston University
ACADEMIC PRESS NEW YORK AND LONDON
COPYRIGHT © 1971, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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Cover design by Jorge Hernandez
FOR JACK AND FLORENCE
PREFACE
There is a well-known bit of advice for an author which says that preferably
before beginning, but certainly before finishing a book, he should ask himself
why he is writing it. This fundamental question and the author's answer to it
are of interest to the potential user of a mathematical text within the context
of a web of related questions. Is this book really necessary? Does it have any
novel features that distinguish it from other books? For whom is it designed?
What are the prerequisites ? Where does it take the reader ? What is the subject
matter ? How is the material organized and structured ?
In recent years it has become fairly standard practice for colleges to offer
a one-semester course in linear algebra at the sophomore level, the justification
and purpose being the connection with certain topics normally covered in
second year calculus. After teaching such a course on a number of occasions,
with singular lack of success, I emerged with the strongly held view that not
only was the justification for the course insufficient, but even more that the
nature of the material was inappropriate for students at the given level of
mathematical experience. Although the available texts were more than ade-
quate, only the most gifted students were in a position to develop successfully,
in the time-span of a single semester, the required combination of geometric
insight, capacity for abstract reasoning, and technique of formal proof. The
conclusion to be drawn, almost inevitably, was that the study of linear algebra
should be deferred, reserving it for students with some mathematical maturity.
ix
χ PREFACE
Once the preceding line of argument is accepted, the problem to be ad-
dressed is how to introduce the interested student to the spirit, style, and
content of modern (some prefer to call it " abstract") algebra. There are many
fine texts of this genre available, but they tend to be aimed at the junior level
and are consequently not quite suitable for our purpose. The true beginner
usually finds that they go too far too fast and are too abstract; he is unable to
assimilate the many new definitions and concepts, and does not possess the
mathematical sophistication to manipulate them in a meaningful way.
Out of such considerations there evolved, over a period of several years,
a book with the following characteristics:
It is designed for a full year course at the freshman or the sophomore level.
Since there are no prerequisites other than familiarity with the standard ele-
mentary topics of high school mathematics-(especially algebra), this course
may be taken before, after, simultaneously with, or even in place of first-year
calculus. Therefore, this text can also be used for well-prepared and motivated
high school seniors.
The style is slow-moving, detailed, quite wordy, and occasionally repetitive,
as dictated by a struggle to provide the student (insofar as space and other
constraints permit) with a book he can read with a minimum of outside help.
In this connection it may be usefully noted that preliminary versions of this
text have been used, and found readable, by such diverse groups as: students
of liberal arts intending to major in mathematics, students in a school of
education preparing to become mathematics specialists, and high school
mathematics teachers (in a NSF institute) anxious to enlarge their mathe-
matical horizons by learning material that impinges upon and enriches the
subject matter in their own curriculum.
No attempt has been made to produce an encyclopedic book, or to throw
masses of material at the student, or to probe deeply. Rather, I have tried to
treat a small number of topics, selected, in the main, for their contribution
toward the larger objective of studying algebraic questions arising out of num-
ber theory, with care, precision of thought and expression, and a certain
amount of thoroughness.
Experience provides ample verification of the assumption that in teaching
mathematics it is pedagogically essential to motivate concepts carefully, and
to proceed from the familiar to the unfamiliar, from the specific to the general,
from the concrete to the abstract. Adherence to these guidelines is a central
feature of this book. Thus, we begin by studying the most familiar mathe-
matical objects, the integers, and then use them to motivate, introduce, and
illustrate some of the basic concepts of modern algebra. The underlying
thread or connective tissue for the book is the interweaving of certain ques-
tions of elementary number theory with the tools, techniques and concepts
of modern, abstract algebra. Topics from linear algebra are consciously ex-
PREFACE xi
eluded since they would constitute a diversion from the main thrust of the
book.
Each section (with the exception of the first few, which are relatively short)
centers on a single topic and develops it rather fully—thus the sections tend
to be considerably longer than is customary. Roughly speaking, the material
in each section is arranged according to increasing complexity and with de-
creasing detail. Many examples are included, mostly of a numerical or com-
putational nature. They serve to illustrate and reinforce the discussion in the
text and to provide previews of coming attractions. Each section closes with
a large number of problems, ranging from the purely mechanical to the
theoretical and abstract. While knowledge of the problems is not needed for
subsequent developments in the text, it is essential that the student work on
lots of them, even if his success is limited. This is the best way really to under-
stand and master the subject matter. Selected answers to the problems appear
at the back of the book. At the end of each chapter, there is a collection of
"miscellaneous problems." These are usually of a more advanced type; some
of them, if expanded fully, could make up an entire section in this, or any
other, book. Only the very best students can be expected to make serious
inroads in the miscellaneous problems.
A few additional comments are in order. I have preferred to plunge right
in and deal with mathematical questions instead of starting with a preliminary
section on sets and functions. These are discussed at the appropriate time—
namely, when they are needed. Some may consider the selection of topics
perverse on the grounds of being too pedestrian or too difficult. I would merely
observe that the amount of space devoted to a topic in the text is not neces-
sarily a measure of its mathematical importance. Moreover, the precise con-
tents, and the emphasis, of a given course are determined according to the
personal tastes of the instructor. This book attempts to give him some leeway
in his choices.
It remains for me to express my deep appreciation to: Sandra Spinacci,
who typed the manuscript with her usual skill and efficiency; Bill Adams, who
read the manuscript carefully and found a number of errors; my family and
S. Shufro, who provided encouragement and assistance; my students, who
participated (unknowingly) in all kinds of pedagogical experiments ; the people
at Academic Press, individually and collectively, for their competence, cour-
tesy, and unfailing cooperation.
I
ELEMENTARY NUMBER THEORY
In this chapter, we shall be concerned exclusively with the set of all integers
—positive, negative, and zero. This set is denoted by
Z={0, ±1, ±2, ±3,...}
and whenever we write any of the symbols a, b, c, d,..., x, y> z,... they shall
represent elements of Z.
It is taken for granted that we are familiar with the integers; however, in
order not to complicate matters unnecessarily at the start, we choose to be
imprecise as to exactly what we know about them. Of course, it is assumed that
we know how to operate, or compute, with integers. As the discussion pro-
gresses, additional rather obvious properties of the integers will be used, and
we shall try to point these up explicitly at the appropriate time. Thus, at the
current stage, our attitude toward the integers will be relatively informal as
compared to the formal axiomatic approach to algebraic objects that will be
adopted in later chapters.
1
2 I. ELEMENTARY NUMBER THEORY
l-l. Divisibility
It is assumed that the facts about addition, subtraction, and multiplication
of integers are known ; however, " officially " we do not recognize the existence
of "division." We consider only objects that belong to Z, so symbols like
" 5/2 " or " 2/5 " are outside our realm of discourse. From this point of view,
"fractions," that is, symbols where me Ζ (we read me Ζ as: m
is an element of Z, m is in Z, or m belongs to Z) and ne Ζ have no meaning.
Our purpose here is to begin the investigation of questions of " divisibility " in
Z, without going outside of Ζ to do so.
1-1-1. Definition. Given a, b e Ζ (this notation asserts that both a and b
are elements of Z) with a Φ0; we say that a divides b (and write this
symbolically as a \ b) when there exists c e Ζ such that ac = b. There are
alternate ways to express this—namely, a is a divisor of b, a is a factor of b,
b is a multiple of a, b is divisible by a.
If no such c exists, we say that a does not divide b (or: a is not a divisor
of b, a is not a factor of b, b is not a multiple of a, b is not divisible by a)
and write ajfb.
It should be noted that the statement a \ b includes the fact that α Φ 0; thus,
according to the definition, 0 does not divide any integer.
The definition of a \ b is concerned only with the theoretical question of
the existence of the desired element c—it says nothing about how to decide, in
practice, if such an integer c exists or how to find it. For example, suppose
we wish to decide if a = 7 divides b = 91. The reader would settle this quickly
by cheating—that is, by using long division. For us, unfortunately, the restric-
tion on division remains in force (in particular, we do not know about long
division), so we must somehow search through Ζ seeking an element c for
which 1c = 91. By testing c = 1, 2, 3,..., in turn, we see eventually that
7-13 = 91, so that 71 91.
In the same spirit, let us decide if a = 1 divides b = 99. Having observed
that 7 · 13 = 91, we note further that
7· 14 = 98 < 99 <7· 15 = 105.
Therefore, if c < 14, then 1c < 99, and if c > 15, then Ίο 99; it follows that
7^99.
The discussion above is designed to emphasize our avoidance of long
division, and surely it would not be very exciting to test other pairs of integers
a and b for divisibility. Instead, we turn to some of the immediate theoretical
consequences of the definition of divisibility.
1-1. DIVISIBILITY 3
1-1-2. Facts. For elements of Z, the following properties hold :
(i) a 10 for every α Φ 0.
Proof'. Here b = 0, so by taking c = 0we have ac = b. |
(ιϊ) If α 16 and ό | c then Û | C.
9
Proof: By hypothesis, there exist dee Ζ such that b = ad and c = èe.
9
Therefore, c = a(de), so that </e is an integer whose existence guarantees that
a\c. I
(HI) (±l)\a for all a, and (±a)|a for all α Φ 0.
iVöö/: Both statements follow immediately from the equation a = (a)(1) =
(-*)(-1). I
(iv) The following four assertions are equivalent:
0) a\b,
(2) a\(-b\
(3) (~a)\b,
(4) (-a)\(-b).
Proof: By equivalence of the four assertions, we mean that any two of
them are equivalent. Of course, in general, two assertions are said to be
equivalent when each one implies the other. Thus, equivalence of our four
assertions amounts to saying that each assertion implies the other three—so
there are a total of 12 implications to be proved. However, by proceeding
" cyclically," it clearly suffices to prove only the following four implications
(1)^(2), (2)=* (3), (3)^(4), (4)^(1).
The proofs of all four of these implications proceed the same way, so we con-
tent ourselves with proving just one of them—namely, (2) => (3); the reader
can easily provide proofs for the three remaining implications.
When (2) is given we have a \ (—b), so there exists ce Ζ such that ac = — b.
By the properties of "minus," we have, therefore, b = ( — a)(c). This says that
( — a) I b and completes the proof that (2) => (3). |
A common notation used to express all the equivalences just proved is
a\b ο a\(-b) ο (-a)\b ο (-a)\(-b)
or simply,
(1) ο (2) ο (3) ο (4).
One immediate, but important, consequence of these equivalences is that
in considering questions of divisibility there is nothing lost in assuming that
a > 0—for after all, a\b ο (-a)\b.
(v) If a I b and a \ c, then a \ (bx + cy) for all choices of x> y e Z. In particu-
lar, if a I b and a \ c, then a \ (b + c) and a \ (b - c).
4 I. ELEMENTARY NUMBER THEORY
Proof: We must show that bx + cy can be written in the form a times an
integer. The proof is straightforward. By hypothesis, we may write b = ad,
c = ae, where d, e ε Ζ. Therefore.
bx + cy = adx + aey = a(dx + ey)
which says that a\(bx +cy). In particular, by making a specific choice of χ
and y—namely, χ = \,y = 1—we see that a\(b + c). Similarly, by putting
χ = 1, y = -1, we see that a\(b — c). |
It is convenient to call any element of form bx +cy (or xb + yc), where
x, y e Z, a linear combination of b and c. Note that the following numbers are
(or more precisely, can be expressed as) linear combinations of b and c—b
(take χ = 1, y =0),-c (take χ = 0, y = -1), 0 (take χ = 0, y = 0), 26 + 3c,
36 - 5c = 36 + (-5)c, -36 + 5c, .... With this terminology, the result
proved above may be restated in the following form : If a divides both b and c,
then it divides any linear combination of b and c; in particular, if a divides b
and c, then it divides their sum and their difference.
(vi) \\ = \b\ ο a = ±b.
a
Proof: Many of us may well be prepared to consider this assertion as
obvious, but a somewhat more convincing argument is required. For this,
it is necessary to recall the definition of " absolute value."
First of all, one defines
|0| =0.
Then, for a Φ0, one notes that the integers a and —a are distinct, with one
of them positive and the other negative. (Naturally, we are assuming that the
basic facts about positive and negative integers are known. It may also be
noted in passing that 0 is, by common usage, neither positive nor negative.)
One then defines, for α Φ 0,
\a\ = the positive member of the set {a, -a}.
Thus, \a\ is either a or —a, and we "abbreviate" this by writing \a\ = ±a. Of
course, this definition of absolute value is the same as the one commonly used
for real numbers; some of the properties of absolute value are given in the
problems—see 1-1-3, Problem 7 .
Returning to the proof, let us suppose that a = ±b—that is, a equals plus
or minus b. The trivial case occurs when a = b = 0, as it is then immediate
that \a\ = |6| = 0. Upon discarding the trivial case, we have both α φ 0 and
b Φ 0. In this case, a = ±b and also — a = ±6; so it follows that {a, —a} =
{b, —b)—meaning that the two sets {a, —a) and {b, —b}, each of which
consists of two distinct elements, are identical. The definition of absolute
value now says that \a\ = \b\— thus proving the implication <=.