Introduction to Higher Algebra

466 Pages·1963·19.78 MB·English

by  Andrzej Mostowski

OTHER TITLES IN THE SERIES
ON PURE AND APPLIED MATHEMATICS
Vol. 1. WALLACE—An Introduction to Algebraic Topology
Vol. 2. PEDOE—Circles
Vol. 3. SPAIN—Analytical Conies
Vol. 4. MIKHLIN—Integral Equations
Vol. 5. EGGLESTON—Problems in Euclidean Space: Application
of Convexity
Vol. 6. WALLACE—Homology Theory on Algebraic Varieties
Vol. 7. NOBLE—Methods Based on the Wiener-Hopf Technique
for the Solution of Partial Differential Equations
Vol. 8. MIKUSINSKI—Operational Calculus
Vol. 9. HEINE—Group Theory in Quantum Mechanics
Vol. 10. BLAND—The Theory of Linear Viscoelasticity
Vol. 11. KURTII—Axiomatics of Classical Statistical Mechanic**
Vol. 12. FUCHS—Abelian Groups
Vol. 13. KURATOWSKI—Introduction to Set Theory and Topology
Vol. 14. SPAIN—Analytical Quadrics
Vol. 15. HARTMAN and MIKUSINSKI—The Theory of Lebesgue
Measure and Integration
Vol. 16. KULCZYCKI—Non-Euclidean Geometry
Vol. 17. KURATOWSKI—Introduction to Calculus
Vol. 18. GERONIMUS—Polynomials Orthogonal on a Circle and
Interval
Vol. 19. ELSGOLC—Calculus of Variations
Vol. 20. ALEXITS—Convergence Problems of Orthogonal Series
Vol. 21. FUCHS and LEVIN—Functions of a Complex Variable,
Volume II
Vol. 22. GOODSTEIN—Fundamental Concepts of Mathematics
Vol. 23. KEENE—Abstract Sets and Finite Ordinals
Vol. 24. DITKIN and PRUDNIKOV—Operational Calculus in Two
Variables and its Applications
Vol. 25. VEKUA—Generalized Analytic Equations
Vol. 26. AMIR-MOÉZ and FASS—Elements of Linear Spaces
Vol. 27. GRADSHTEIN—Direct and Converse Theorems
Vol. 28. FUCHS—Partially Ordered Algebraic Systems
Vol. 29. POSTNIKOV—Functions of Galois Theory
Vol. 30. BERMANT—A Course of Mathematical Analysis Part
Vol. 31. LUKASIEWICZ—Elements of Mathematical Logic
Vol. 32. VULIKH—Introduction to Functional Analysis for Scien­
tists and Technologists
Vol. 33. PEDOE—An Introduction to Protective Geometry
Vol. 34. TIMAN—Theory of Approximation of Functions of
a Real Variable
Vol. 35. CSASZAR—Foundations of General Topology
Vol. 36. BRONSHTEIN and SEMENDTAYEV—A Guide-Book to
Mathematics for Technologists and Engineers
INTRODUCTION
TO HIGHER ALGEBRA
by
A. MOSTOWSKI and M. STARK
Member of the Polish Institute of Mathematica
Academy of Sciences of the Polish Academy
of Sciences
TRANSLATED FROM THE POLISH
by
Dr. J. MUSIELAK
University of Poznan
PEEGAMON PEESS
OXFORD . LONDON · NEW YORK · PARIS
PWN—POLISH SCIENTIFIC PUBLISHEES
WARSZAWA
1964
PERGAMON PRESS LTD.
Ileadington UHI Eoli, Oxford
4 and 5 Fitzroy Square, London W.l.
PERGAMON PRESS INC.
122 East 55th Street, New York 22, N.T.
GAUTHIER-VILLARS ED.
55 Quai des Grands-Augustins, Paris, 6e
PERGAMON PRESS G.m.b.H.
Kaiserstrasse 75, Frankfurt am Main
Distributed in the Western Hemisphere by
THE MACMILLAN COMPANY . NEW YORK
pursuant to a special arrangement with
Pergamon Press Limited
Copyright Q 1964 by
PASSTWOWE WYDAWNICTWO NAUKOWE
(PWN—POLISH SCIENTIFIC PUBLISHERS)
WARSZAWA
This book is a translation of the original Polish
Elementy algebry wyzszej
published by PWN—Polish Scientific Publishers 1958
Library of Congress Catalogue Card Number 63-11923
Printed in Poland (D.Ü.J.)
CHAPTER I
INTRODUCTION
§ 1. FUNCTIONS
1. Sets
Mathematical investigations are based on the notion
of a set. We shall not define what we mean by a set;
we give only a few examples:
The set of inhabitants of Great Britain at January 1,
1920;
the set of atoms constituting a given substance;
the set of even numbers;
the set of points of a fixed plane.
Objects the totality of which constitute the set are
called elements of the set. Instead of writing x is an element
of the set X we write
9
x e X;
if x is not an element of the set X, we write
x4 X.
If every element of a set X is also an element of
a set Y, then we call X a subset of the set Y or we say
that the set X is contained in the set Y and we write
XC Y.
E.g., the set of even numbers is contained in the set
of all integers, the set of points of a circle is contained
in the set of all points of the plane.
If every element of a set X belongs to the set Y and
every element of the set Y belongs to the set X, then
the sets X and Y are identical.
12 INTRODUCTION TO HIGHER ALGEBRA
It, follows from the definition that every set is its own
subsety i.e.
I C I.
From the examples given above we see that elements
of a set can be of various kinds. They can be objects
of the real world as well as abstract notions.
In mathematics it is useful to consider a further set,
differing from the examples given above, the so-called
empty set or void set which contains no element. For
instance, instead of saying "there exists no even divisor
of 15" one can say "the set of even divisors of the number
15 is empty". This will enable us to formulate many theo­
rems and proofs in a simpler way. The empty set is
a subset of any set.
2. Functions
The notion of functions is well known e.g. from
various physical laws. The length of the path in a uni­
formly accelerated motion is a function of the initial
velocity, of the time and of the acceleration: s = v t
0
+ %at\
The intensity of electric current is the following
function of the voltage and resistance: J = E/R.
The unit of length being fixed, the area of a circle
is a function of one variable, namely, of the length of
the radius: P = nr2.
All these examples of functions are special cases
of the most general definition which will be given now.
Let X and Y be two sets. Assume that with every
element x of the set X we associate an element of the
set Y and that every element of the set Y is associated
with an element of the set X. Such a correspondence
is called a function. The element of the set Y associated,
by a function /, with the element x of the set X is denoted
by the symbol f(x) and is called the value of the function f
at x. X is the set of arguments (the domain) of the function
and Y is the set of values (the range) of the function.
INTRODUCTION 13
In the first of the above given examples, the domain
of the function consists of triples of numbers v , t, a,
0
and the range consists of numbers.
In the second example, the domain consists of pairs
of numbers E, R, and the range—of numbers.
In the third example, the domain as well as the range
consists of numbers.
Consider another example of a function. Enumerate
five chairs and place five persons on these five chairs.
Associate with each of numbers 7 = 1,2,3,4,5 the
person sitting on the jih chair. We obtain a function
whose arguments are numbers 1,2,3,4,5, and values
are persons.
Associate with every point of a circle, the other end-
point of the diameter passing through this point; a function
is obtained whose domain and range consist of points
of the same circle.
In mathematics we mostly consider functions whose
arguments are numbers or pairs of numbers, triples of
numbers etc., and whose values are numbers, for example
f(x) = x2, f(œ,y) = œ2 + y2, f{x,y,z) = x + y + z.
Such functions are called functions of one variable,
of two variables, of three variables etc., respectively.
In place of the term function it is sometimes more
convenient to use the word transformation-, we shall use
this term many times.
In the sequel we shall often denote a function by
the symbol /. The symbol f{x) is also often used to
denote a function. However, the last notation is equivocal,
for f(x) can denote the function as well as its value at
the argument x. In cases where no misunderstanding is
expected to arise, we shall use the symbol f(x). However,
if any doubt might appear, we shall use the terminology
"the function /(a?)" or briefly, /.
Ambiguity of this type arises often in the case of
constant functions. Let e.g. f(x) = 3 for all numbers x.
14 INTRODUCTION TO HIGHER ALGEBRA
Writing "3", we might have in mind the number 3 as
well as the constant function: f(x) = 3 for all œ. Not­
withstanding we shall use the sign "c" to denote the
constant function f(x) = c for all x\ of course, we shall
be careful in applying this notation. The fact that f(x)
is a constant function will often be denoted by f(x)
= const or / = const.
Two functions / and g defined on a set are equal, if
/(#) = g(x) for every element x e X. For example, func­
tions f(x) = (x—l)2 and g(x) = cfi—2x+l are equal, since
(x—l)2 = x2—2x+l for every number x.
3. Operations on functions
Suppose that we are given two functions f and /
x a
defined on a set X and assuming numerical values. Then
we can form new functions gr, h and k defined by means
of formulae
g{x)=Hx)+Ux),
h(x)=f(x)-f (x), for x e X.
l 2
Tc{x)=U{x)Ux),
If, moreover, /,(») φ 0 for all x e X, then we can form
a function I defined by
·<·>-!$·
These functions will be denoted
Λ + /2* /l""/2> /1/2 > / ?
/2
respectively; in the last case we shall also use the equiv­
alent notation
Mi-
Thus, according to the definition of the operations on
functions, the following equations hold:
(h+/,) (χ) = h(x)+h(x), (A/ ) (x) = h(x) Ux),
2
(u-tM*)=m-m, £(») =£@.
/2 /2W
INTRODUCTION 15
In this manner we defined four operations on functions:
addition, subtraction, multiplication and division.
E.g., denoting the trigonometric functions defined in
interval 0 < x < |π by sin, cos, tan, cot, we may write
cos. tan = sin, sin. cot = cos .
If c denotes the constant function assuming the
value c everywhere, then the function cf assumes the
value c.f(x) for every x.
The above defined operations on functions have the
following properties:
Commutativity of addition and multiplication:
/l + /« = /2 + /l, /l/ = //l.
2 2
Associativity of addition and multiplication:
h+(/.+/.) = (h+/.)+h, UUU) = (fiUU ·
Distributivity of multiplication with respect to addition
and subtraction:
HU+U) = Uh+hh, MU - h) = nn-hU ·
Further properties:
If fi + h = hi then h = h~~h\ we a^so ^have ^e equation
fi , h hU + hh
=
/2 /4 /2/4
and many others, analogous to those for operations on
numbers.
To prove these properties, it is sufficient to verify
that the function on the left-hand side of the equation
assumes, for every value of x, the same value as the
function on the right-hand side of the equation.
Functions a,bx, ex2,..., kxn, ... are called monomials in
the variable x. The sum of a finite number of monomials
in one variable is called a polynomial in one variable.
Every polynomial in one variable is of the form
(1) φ = α χη + α χη-1 + ... + α .
0 1 η
16 INTRODUCTION TO HIGHER ALGEBRA
If a Φ 0 and n > 1, then the positive integer n is called
0
the degree of the polynomial (1) and will be denoted by
ag(p; if φ = const, then we shall write dgç> = 0.
A monomial in two variables x , x is a function of
x 2
the form
αχ\χ\, α^Ο.
The number Tc + l is called the degree of this monomial;
here we assume a?5 = 1 and dg const = 0. A sum of a finite
number of monomials is called a polynomial in variables
x ,x \ the highest of degrees of terms which remain
x 2
after reduction of similar terms is called the degree of
the polynomial.
Polynomials in three and more variables will be
defined in a similar manner. For instance, if
φ{x , x , x, x ) = x\ + 3x1 + 2x x\ + 3x x + #i# # # ,
x 2 z à x ± A 2 3 4
then dgç? = 4.
It follows from the definition of a polynomial that
a sum, a difference and a product of two polynomials are
polynomials.
However, it may be proved that a quotient of two
polynomials is not always a polynomial, e.g. IIx.
A quotient of two polynomials is called a rational
function.
§ 2. MATHEMATICAL INDUCTION
1. Positive integers and the mathematical induction
The numbers 1, 2, 3, ... are called positive integers or
natural numbers) we shall denote the set of these numbers
by the symbol N.
The following theorem which we shall accept without
proof gives a fundamental property of the set N.
THEOREM 1. In every non-empty set X made up of
positive integers there exists the least number.
INTRODUCTION 17
Denote this number by n\ thus, it has the following
properties:
(i) n e l;
(ii) m < n implies mj X.
For instance, the number 1 is the least number in
the set N itself; the number 2 is the least even number,
i.e. the least number in the set of even numbers. It can
be proved that the number 1729 is the least number
which can be expressed in two different ways as a sum of
two cubes of positive integers (1729 = 123+13 = 103 + 93).
Theorem 1 implies theorem 2.
THEOREM 2 (principle of mathematical induc­
tion). If X is a set of numbers satisfying the two following
conditions:
a. 1 belongs to the set X;
b. if n belongs to X, then n+1 belongs to X,
then the set X contains all positive integers.
The principle of mathematical induction can be illustrated by
the following example.
Assume that we have infinitely many dominoes standing in
a row, one after another. If one stone falls, then the next stone
falls, too. Thus, if the first stone falls then, one after another, all
stones will fall.
Proof. Suppose there exist positive integers not
belonging to X and let Z be the set of all such numbers.
As follows from theorem 1, the set Z contains the least
number n . This number cannot be equal to 1, for 1 be­
0
longs to the set X. Therefore the number Z = n —1 is
0
a positive integer and since it is less than n it does not
0J
belong to Z, i.e. it belongs to X. Hence it follows from
the second condition that l+l = n belongs to X, in
0
contradiction to the supposition that n belongs to Z.
0
Thus, the supposition of existence of positive integers
not belonging to X leads to a contradiction.
In general, theorem 2 is used in another formulation.
In order to give this formulation we introduce the follow­
ing notation.
2