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