A. I Kypour

AJII ESPAMtIECKHE YPABHEHNSI

IIPON3BOJlbHbIX CTEl1EHEA

H3gRTe.1bCTBO «Hayrca»

A.U. Kurosh

ALGE B RAIC

EQUATIONSOF ARBITRARY

DEGREES

Translated from the Russian

byV. Kisin

t erst punhsned I/li

Revised from the l975 Russian edition

HQ OHZABQCKO.u A3slKC

! English translation, Mir Publishers, 1977

t 'ontents

7

9

10In1.

2.

3.

te

19

6.

7.

Conclusion

Bibliography

Preface

troduction

Complex Numbers

Evolution. Quadratic Equations

Cubic Equations

Solution of Equations in Terms of Radicals and the Existenceof Roots of Equations

The Number of Real Roots

Approximate Solution of Equations

Fields

22

24

27

30

35

36

IIUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUI

Prefact

the author's lecture to highMathematics Olympiad at

a review of the results andalgebraic equations with due>f its readers. No proofs arehave required copying almost,her algebra. Despite such aneke for light reading. Even air the reader's concentration,initions and statements, check, application of the methods

This booklet is a revision ofschool students taking part in thiMoscow State University. It givesmethods of the general theory ofregard for the level of knowledge iincluded in the text since this wouldhalf of a university textbook on hi~approach, this booklet does not m:popular mathematics book calls fcthorough consideration of all the delof calculations in all the exatnples,described to his own examples, etc.

IIUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUURRRRRRRRRRRRRRRRRRRRRRRRRRRURRURRRRRRUUUUUI

Introduction

A secondary school course of algebra is diversified but equationsits focus. Let us restrict ourselves to equations with one

tnown, and recall what is taught in secondary. school.Any pupil can solve jirst degree equations: if an equation

ax+ b =0

areunl

~Jven, in which a 4 0, then its single root is is

X =a

Furthermore, a pupil knows the formula l' or solving quadratic<ations eqt

ax'+ bx+ c = 0

ere a/0: wh

� 5+ rb' 4ac-x =

2a

i <l III

tations. For example, the third-degree cubic! equation

ax + bx + ex=0

ich has one root x = 0, and. after factoring out x, is transformeda a quadratic equation

ax +bx+c=O

whinti

n

A fourth-d

1. Complex Numbers

x2+ 1 =0

is the simplesrealm of nun

In his st<the system ofstarts with inn

c l........

legree quartic! equation

ay +by +c=0

atic, may also be reduced to a quadratic equation= x, calculating the roots of the resulting quadraticthen extracting their square roots.phasize once again that these are only very specialc and quartic equations. Secondary school algebralods of solving arbitrary equations of these degrees,Itore so of higher degrees. However, we encounterdgebraic equations in different branches of engineering,9 physics. The theory of algebraic equations of anee n, where n is a positive integer, has requiredlevelop and now constitutes one of the main partsbra taught at universities and pedagogical institutes.

' of algebraic equations is essentially based on themplex numbers taught at high school. However,a doubt the justification for introducing these

their actual existence. When complex numbersced, even mathematicians doubted their actual>ce the term "imaginary numbers" which stillvever, modern science sees nothing mysterious

cc '

t of such equations. Is there a way to expand thetbers so that these equations also possess roots?!dy of mathematics at school the student sees

numbers at his disposal constantly extended. He.grai positive numbers in elementary arithmetic. Very

.........1. e r...... s. ne. gati.ve..~umber'...a .i. J

called biquadrby setting y~equation and

Let us emtypes of cubigives no met?and all the rlhigher degree tmechanics aniarbitrary degrcenturies to cof higher alger

The theorytheory of co:students ofte;numbers andwere introduexistence, he>survives. Hoi

d, the equation

mbers, and the

m the way toed to a general

ted on a givenunit of scale iscan be put in

0 1

iIG

l number whichnce between Aor the distance

ibers in such ast as natural anot constructedobers.which this newhat objects will

to carry out

previously had no roots. Thus, the equation

2x � 1 = 0

acquires a root only after fractions are introducex+1=0

has a root after the introduction of negative nuequation

x~ � 2=0

has a root only after irrational numbers are addeAll this completely justifies one more step

enlarge the store of numbers. We shall now proceoutline of this last step.

It is known that if a positive direction is fi,straight line, if the origin 0 is marked, and if achosen Fig. 1!, then each point A on this line

correspondence with its coordinate, i. e. with a reaexpresses in the chosen units of scale the distaand 0 if A lies to the right of the point 0,

'

~ Ii

But is it possible to expand the store of nunway that new numbers can be represented in jumanner by the points of a plane? So far we havea system of numbers wider than that of real nur

We shall start by indicating the "material" withsystem of numbers is to be "constructed", i. e. wact as new numbers. We must also define hov

FIG. 2

I I I I I ]

U U I I U U

U

implest way is to consider the points ofthe new numbers. To seriously conyiderwe must merely define how to carry outi them, i. e. which point is to be theof the plane, which one is to be their

I' a point on a straight line is completelyt number, its coordinate, the positiona plane can be defined by a pair of realus take two perpendicular straight lines,

at the point 0 and on each of themand set olf a unit of scale Fig. 2!. Let

two positive real numbers, e. g. t ethe selected units of scale the distance

Irdinate axis the abscissa of point A!,gives in the selected units of scale its

sa axis the ordinate of the point A!. a, b! of positive real numbers we can

ly defined point in the first quadrantU

points of the plane, the sthe plane themselves asthese points as nuinbers,algebraic operations wittsum of two given pointsproduct, etc.

Just as the position oldefined by a single reaof an arbitrary point onnumbers. To do this letintersecting on the planefix the positive direction

is completely defined bynumber a which gives infrom this point to the cand the number b whichdistance from the abscis,Conversely, for each pairindicate a single precise'.

toeralS lrlantsaxisthe

d b

byif aoint

iintsnely

to!ef s.lntilduct:is sa

a, b! + c, d! = a + c, b + d!

Dn the other hand, let us call the product of the given pithe point with the abscissa ac � bd and the ordinate ad + bc,

>intse.

be interchanged!, associative i. e. the sum and the producthree points are independent of the position of the brackets!distributive i. e. the brackets can be removed!. Note that theof association for addition and multiplication of points maktpossible to introduce in an unambiguous way the sum andproduct of any finite number of points on the plane.

Now we can also perform the operations of subtractionn n

t ofandlaw

:s itthe

and

plane. and the pairs of coordinates a, b!, i. e. in orderavoid the same pair of coordinates a, b! corresponding to sevdistinct points on the plane, we assume the abscissas of pointquadrants 11 and 111 and the ordinates of points in quadrIII and IV to be negative. Note that points on the abscissaare given by coordinates of the type a, 0!, and those onordinate axis by coordinates of the type �, b!, where a anare certain real numbers,

We are now able to define all the points on the planepairs of real numbers. This enables us to talk further not <point A, given by the coordinates a, b!, but simply of a p a, b!.

Let us now define addition and multiplication of the pcon the plane. At first these definitions may seem extrerartificial. However, only such definitions will make it possibhrealize our goal of taking square roots of negative real nuinl

Let the points a, b! and c, d! be given on the plane. l.now we did not know how to define the sum and the proiof these points. Let us call their sum the point with the ahaa+ c and the ordinate b+ d, e.

a, 6! ac+ bd bc � ad ! c,d! c+d' c+d J

fie abscissa axis,

a, 0! b, 0! = ab, 0!

duce to additiond for subtraction

�, 0! 6'

i axis re resents

in 'll IIIII~ I II IIIIIIIIIlIIIIIIIJIIIIII'IIIIIIIthe point �, 1!i. Let us denote

plication of the

, defined above!equality by theis even simpler

ht-hand side ofd equal to the

by the divisor yields the dividend!. Thus, a, 6! � c, d! = a � c, b � d!

The reader will easily see that the product asof the point on the right-hand side of the last

appoint c, d! is indeed equal to the point a, 6!. Itto verify that the sum of the point on the rigthe first equality and the point c, d! is indeepoint a, 6!.

By applying our definitions to the points on t.'e. to the points of the type a, 0!, we obtain.

a, 0! + �, 0! = a + 6, 0!

i. e. addition and multiplication of these points reiand multiplication of their abscissas. The same is valiand division:

a, 0! � b, 0! = a � 6, 0!

If we assume that each oint a 0 of the abscissr

with real numbers. For example, let us considerwhich lies at a distance 1 upward of the point Cthis point by the letter i:

i = �, 1!

and let us find its square in the sense of multipoints on the plane:

axis, not on the� 1, i. e.

a, b! = a, 0!+ �, b!

therefore a real.ation, the second

numbers we can>ns with complex

c!i

imbers a numbere. we can now

>is root is givennt �, 1!, whichi the plane, and.maginary unity"ing on the plane.tructed is more:d the system ofith the operationss not difficult toby real numbersFor example, letof addition, the

However, the point � 1, 0! lies on the abscissaordinate axis, and thus re'presents a real nuinber

i~ = � 1

Hence, we have found in our new system of mwhose square is equal to a real number � 1, i.find the square root of � 1. Another value of tlby the point � i = �, � 1!. Note that the poiwe denoted as i, is a precisely defined point or.the fact that it is usually referred to as an "Idoes not in the least prevent it from actually exist

The system of numbers we have just consextensive than that of real numbers and is call complex ppmbers. The points on the plane together wwe have defined are called complex numbers. It iprove that any complex number can be expressedand the nuinber i by means of these operations:us take point a, b!. By virtue of the definitionfollowing is valid:

The addend a, 0! lies on the abscissa axis and isnumber a. By virtue of the definition of rnultiplic;addend can be written in the form

�, b! = b, 0!�, I!

1 <], ' I i 1 1] I ~ ll ~ ~

By means of this andard notation of compleximmediately rewrite t e above formulas for operatii'.numbers:

a + bi! + c + di! = a + c! + b + 4

a+ bi! c+ di! = ac � bd! + ad + $

a bi � c+ i'I = � c!+

2. Evolution.

Quadratic Equations

can extract squareany negative real' � a is a negative

V- =+v

>are root of the

:quation with real 0, this equation

complex numbers,

n of multiplicationnent with the law:cond of the aboverule of binomial

w of distribution!,luce similar terms,de of the second

b is taken in both

any a and b both:oefficient in i willpossesses two values~ to the following

radical is added:gative one to the

It should be noted that the above definitioof the points of the plane is in perfect agreerof distribution: if on the left-hand side of the @equations we calculate the product by themultiplication which itself stems from the laand then apply the equality i~ = � I and redwe shall arrive precisely at the right-hand sIequation.

Having complex numbers at our disposal, weroots not only of the number � l, but ofnumber, always obtaming two distinct values. IIreal number, i, e. a > 0, then

where Pa is the positive value of the sqipositive number a.

Returning to the solution of the quadratic icoefficients, we can now say that where b2 � 4acalso has two distinct roots, this time complex.

Now we are able to take square roots of any

rim '~immi Iia s nlIwhere the positive value of the radical Pa +terms. Of course, the reader will see that for;the first term on the right-hand side and the cbe real numbers. Each of these two radicals pcwhich are combined with each other accordinlrule: if b > 0, then the positive value of oneto the positive value of the' other, and the ne

xg =

11, � � +

2

ar equation are the numbers

: 3+i, xg �� 1 � 2i

d that each of these numbers indeed

the problem of extracting roots of anindex n from complex numbers. It can

:omplex number m there exist exactly n> such that raised to the power n i. e.n factors equal to this number!, eachother words, the following extremely

I

>f any complex number has exactly n

y applicable to real numbers, which aretplex numbers: the nth root of a realdistinct values which in a general case

that among these values 'there will beers, depending on the sign of the number< n.f one has three values:

y~l and � � � i�2 2 2

the formula for taking the square root+ bi, This formula reduces calculationquare roots of two positive real numbers.no formula exists which would expressx number a+ bi in tertns of real valuesiliary real numbers; it was proved thatbe:derived. Roots of order n of cotnplex

Therefore, the roots of oi

It can easily be checkesatisfies the equation.

I.et us now turn toarbitrary positive integralbe proved that for any <distinct complex number<if we take a product ofyields the number a. Initn ortant theorem holds.

A root of order ndistinct cont lex values.

This t eorem is equaHa particular case of connumber a has precisely nare complex. We knowtwo, one or no real numba and parity of the inde>

Thus, the cube root c

In section 1 we gaveof a complex number aof the root to extr'acting e

@Unfortunately, for n o 2the nth root of a compieof radicals of certain auxno such formula can ever

g3.Cubic Equations

- ax + bx + c = 0 x

iis equation, setting

ax=y ��

3

written as

'+ + =0

+ p +' 27+

ig quadratic equations is also valid foriird-degree equations, usually called cubicerive a formula, which, although moreh radicals the roots of these equations'his formula is also valid for equationsefficients.

a. Substituting this expression of x intocubic equation with respect to y, whichcient of y~ will be zero. The coefficientl the absolute term will be, respectively,

2a abtb> il= � � +c

27 3

the three cube radicals has three values.iot be combined in an arbitrary manner.

The formula for solviticomplex coefficients. For thequations, we can also dcomplicated, expresses witin terms of coefficients. 'Iwith arbitrary complex co

Let an equation

be given. We transform tl

where y is a new unknowiour equation, we obtain ais simpler, since the coeffiof the lirst power of y anithe numbers

i. e. the equation can be

We know that each ofHowever, these values cani

has three distinct rea

ation

x � 19x+ 30 = 0

i. e. the result is neg:formula yields

33

P27

I' IINSIS

se two values of the radicals must be added

root of the equation. Thus we obtain theuation. Therefore, each cubic equation withhas three roots, which in a general case aremme of these roots may coincide, i, e, constitute

iificance of the above formula is extremely: coefficients p and q be real numbers. Itthe equation

y +py+ a=0

I roots, then the expression

� +4 27

:e the expression is under the square rootextracting this root will yield a complex

!f the two cube root signs. We mentionedof cube roots of complex numbers requires

i, but this can be done only approxiinately,

itive. The first of the cube radicals in the

the number � �. TheP 3together to obtain athree roots of our eqnumerical coefficientscomplex; obviously sca multiple root,

The practical sigrsmall. Indeed, let th<can be shown that if

will be negative. Sinisign in the formula,number under each <above that extractiontrigonometric notatioiby means of tables.

Example. The equ

this formula. But a direct verification demonstrates that these rootsare the integers 2, 3 and � S.

In practice the above formula for solving cubic equations yields

the roots of equations only when the expression � + � is po-4 27

sitive or equal to zero. In the first instance the equation has onereal and two complex roots; in the second instance aH the rootsare real but one of thetn is multiple.

Kxatnple. We want to solve the cubic equation

x' � 9x' + 36x � 80 = 0

Settingx=y+3

we obtain the "reduced" equation

ys+ 9y � 26 =0

Applying the formula, we obtain

P4 27

+ = 196 = 14'

and therefore3

IIIIIIIIIIIIIEIIIITlfffmfftlPlfltt Itfff fffftfffff IIvalue of the second radical will be the number � 1, and one ofthe roots of the reduced equation is

yi =3+ � 1! =2

Knowing one of the roots of the cubic equation, we canobtain the other two in many different ways. For instance, we

� 1+ <+12 and � 1 � i+12

Therefore, the roots of the original cubic equation are the numbers

5, 2+ i+12 and 2 � i~12

Of course, calculation of radicals is not always as easy asin the carefully selected example discussed above; much more oftenthey have to be calculated approximately, yielding only approximatevalues of the roots of an equation.

4. Solution of Equationsin Terms of Radicals

and the Existence of Roots

of Equations

Quartic equations also allow a formula to be worked outwhich expresses the roots of these equations in terms of theircoefficients. Involving still more "multi-storied" radicals, this formula

I] ] l ] ] i I] I ] ' ] ] '1i] ] ' i' ]t

l lhigher degrees. Note that a general form of an equation of degreen, where n is a positive integer, is

aox" + a,x" '+ a,x" + + a�,x+ a�=O

The search continued unsuccessfully until the beginning of the19th century, when the followin s ectacular result was roved:

values of the second radical and add up the mutually correspond.-ing values of the radicals. Or we may divide the left-hand sideof the reduced equation by y � 2, after which we only need tosolve a quadratic equation. Either of these methods will demonstratethat the other two roots of our reduced equation are the numbers

Iequawhoif this tF

x' � 4x � 2 =0

but,pion~

concwherapprthesenn%q ~ ~ ~ i

n addition, for any n, greater than or equal to five, anition can be written of degree n with integral coefficients,se roots are not expressible in radicals, however complicated,e radicands involve only integral or fractional numbers. Suchie equation

t can be proved that this equation has five roots, three realtwo coinplex, but none of these roots can be expressed in:als, i. e. this equation is "unsolvable in terms of radicals-.efore, the store of numbers, both real and complex, whichthe roots of equations with integral coefficients such numbers

called algebraic as opposed to transcendent numbers whichnot the roots of any equations with integral coefficients!, isi greater than that of the numbers which can be written ins of radicals.'he theory of algebraic numbers is an important branch of algebra;ian mathematicians E. I. Zolotarev �847-1878!, G.,F. Voronoi!-1908!, N. G. Chebotarev �894-1947! inade valuable contribu-

in this field.,bel �802-1829! proved that deriving general formulas forng equations of degrees n o 5 in terms of radicals wasissible. Galois �811-1832! demonstrated the existence oftions with integral coefficients, unsolvable in terms of radicals.ilso found the conditions under which the equation can be

shall only mention that presently Soviet mathematicians are:ering in the development of group theory..s far as practical determination of roots of equations iserned, the absence of formulas for solving nth degree equationse n! 5 causes no serious difficulties. Numerous methods ofoximate solution of equations suffice, and even for cubic equations, methods are much auicker than th li 'o or

Iiandradi Therformareare imuclterm

'I

Russ�86ltions

solviimpcequaHe

If there were equations with numerical coefficients, eitheror complex, which possessed no real or complex roots, the sof real numbers would have to be extended. However, thiunnecessary since cotnplex numbers are sufhcient to solveequation with numerical coefficients. The following theorem h<

Any equation of degree n with any numerical coefficients has n rcomplex or, in certain cases, real; some of these roots may coini. e. form multiple roots.

This theorem is called the basic theorem of higher alg~It was proved by D'Alembert �717-1783! and Gauss �777-1as early as the 18th century, although these proofs were perfeto complete rigorousness only in the 19th century; at prethere exist several dozen dilferent proofs of this theorem.

The concept of a multiple root, mentioned in the basic theomeans the following. It can be proved that if an nth deequation

realtores 1sany

!lds:oots>cide,

.bra.855!cted,sent

rem,gree

aox" + a1x" '+ +a�1x+ a�=0

has n roots tx�nz, ..., |x�, then the left-hand side of the equ;can be factored in the following manner:

ition

.Ja,x" + a,x" '+ + a�,x + a�= ao x � m,! x � mq! ... x � 1

sideoots

Conversely, if such a factorization is given for the left-handof our e uation the numbers t«c« ... will be the r

/5. The Number of Real Roots

ionsfor

ems:nts.

The basic theorem of higher algebra has important applicatin theoretical research, but it provides no practical methodsolving the roots of equations. However, many technical prob]require information about the roots of equations with real coeffich

Let an nth

and m=2, the

Let us plolchoose the cocselected for x

I ]I I I I I I III I I I I I I I »]I ql I I I I I i

s I I I I I ]I I

degree equation be given

uox" + a>x" ' + + a� ix + a�= 0

efficients, We already know that it has n roots.m real roots? If so, how many and approximatelylocated? We can answer these questions as follows.he polynoinial on the left-hand side of our equation

x! = aox" + a,x" + + a�,x+ a�

iiliar with the concept of function will understandhe left-hand side of the equation as a function ofTaking for x an arbitrary numerical value m and

into the expression for f x!, after performing all thearrive at a certain number which is called the valueiial f x! and is denoted as f cx!. Thus, if

f x! = x' � Sx' + 2x + 1

f�! = 2 � 5 2 + 2. 2+ 1 = � 7

: a graph of the polynomial f x!, To do this weordinate axes on the plane see above! and, havingi value Ix and calculated a corres ondin value m

:ly, since there are an infinite number of the values>t hope to lind the points n, f u!! for all of themsatisfied with a finite number of points. For theity we can first select several positive and negativeof »x in succession, mark on the plane the pointsto them and then draw through them as smooth

having real ccAre any of thewhere are theyLet us denote tby f x!, i. e.

The reader fanthat we treat tthe variable x.substituting it ioperations, weof the polynon

question.Unfortunat»

of »x one cann»and must be:sake of simplicintegral valuescorresponding

and [a~ = � a for a < 0! and A is the greatest of the absolutevalues of all the other coefficients a,, a2, a�cr�, then

AB= +1

I ~0 I

However, it is often apparent that these bounds are too wide.Example. Plot a graph of the polynomial

f x! = x � 5x m 2x + 1

Here ao! = 1, A = 5, and thus B = 6. Actually, for this particularexample we can restrict ourselves to only those values of m, whichfall between � 1 and 5. Let us compile a table of values of thepolynomial f x! and plot a graph Fig. 3!.

The graph demonstrates that all the three roots a,, e, and a3

i ! li i! 1

1 I I ! 1 I

neighbouring values of a for which the numbers f m! have oppositesigns, and thus it was sufficient just to look ht the table ofvalues of f a!.

If in our example we found less than three points of intersectionof the graph with the abscissa axis, we might think that owingto the imperfection of our graph we traced the curve knowingonly seven of its points!, we could overlook several additional

x' � 8x +x � 2=0

x � 5x +2x+ I=I0

If14

cated between any giventhods will not be stated

se neighbouring integers,tion

e roots of this equation:xample, let us take the

equation and even the number of roots lonumbers a and b, where a b, These mehere.

Sometimes the following theoremsare useful since they give some in-formation on the existence of realand even positive roots.

Any equation of an odd degreewith real coefftcients has at least onereal root.

If the leading. coefftcient ao andthe absolute term a� in an equationwith real coefftcients have oppositesigns, the equation has at least onepositive root. In addition, if our equationis of an even degree, it also has at leastone negative root.

Thus, the equation

has at least one positive root, while.the equation

-x +2x' � x +7x � 1=0

In the previous section we found tho,between which the real roots of the equa

are located. The same method allows thto be found with greater accuracy. For <

f x! = x � 5x + 2x + 1

re difFerent,

n is given,

bf a! � af !f a! - f b!

i! and f b!

sive values>scissa axis,f one-tenth.root u2 toieoretically,h involveslly unman-methods of:ions much:thods and

of theFul to find>ady know,oot to thelues of the

0.9, we can find between which two of these succesof x the graph of the polynomial f x! intersects the af:i. e. we can now calculate the root u2 to the accuracy o

Proceeding further, we can find the value of thethe accuracy of one-hundredth, one-thousandth or, tlto any accuracy we want. However, this approaccumbersome calculations which soon becoine practicaageable. This has led to the development of various .calculating approximate values of real roots of equaiquicker. Below we present the simplest of these miimmediately apply it to the calculation of the rootcubic equation considered above. But first it is use:bounds for this root narrower than the ones we aln0 < m, < 1. For this purpose we shall calculate our raccuracy of one-tenth. If the reader calculates the valpolynomial

for x = 0.1; 0.2; '0.9, he will obtain

f�.7! = 0.293, f�.8! = � 0.088

and therefore, since the signs of these values of f x! a:

0.7 < >x, < 0.8

The method is as follows. An equation of degree

The bound c is calculated by means of the formul,

In this case a =0.7, b = 0.8, and the values of f iare given above. Therefore

iction of a new:re; in essencelied differential

zn-sx+ an

x! 3xz 10x + 2

f" x! = 6x � 10

as f'{x!. Thisule: each term

the exponentnity; moreover,that a�= a�xo.

ynomial f ' x!.h is called thevoted as f" x!.

+2x+ 1 we

close to eachthe same sign

I f b! will be1 f a! are thei, i. e. the onesigns of f" b!bound b, must

The formula for the bound d requires the introdvconcept which will play only an auxiliary role h<it belongs to a dillerent branch of mathematics caicalculus.

Let a polynomial of degree n be given

f x! = aox" +a,x" '+ azx" + + a�zx +i

Let us call the polynomial of degree n � 1!

f'4! = nnvx" ' + in � 1! a,x" + [n � 21 azx'+ 2a�-zx+ a -z

a derivative of this polynomial, and denote itpolynomial is derived from j x! by the following ra�x" ' of the polynomial f x! is multiplied byn � k of x, while the exponent itself is reduced by u:the absolute term a, disappears, since we can consider

We can again take the derivative of the polThis will be a polynomial of degree n � 2!, whic'.second derivative of the polynomial f x! and is de>

Thus, for the above polynomial f x! = x' � 5xobtain

lIIII JlllRllllll,llJill lllTIIB I �IIllIIchosen. If the bounds a, b are chosen sufftcientlyother, the second derivative f" x! will usually havefor x = a and x = b, while the signs of f a! amdifferent, as we know, If the signs of f" a! amsame, d must be calculated with the first formulain which the bound a is used, however if theand f b! coincide, the second formula, involving the

and f b! negative, the ~used. Since f'�.8! = � <

second formula for the bound d must bel,08, we obtain

0.088 = 0.8 � 0.0215 ... = 0.7784 ... d =0.8 ��

x2 we have found the following bounds,: knew before:

Thus for the rootnarrower than those wi

or, if we widen these b

it if we take for ei~ the arithmetic mean,e calculated bounds,

It follows, therefore, th;i. e. half the sum, of th

u = 0.7777

ed 0.0008, equal to half the difference of

III ' III IIII III IIIIits of algebraic equations, which we have The problem of rocove, can be considered in more generalust introduce one of the most important

h

7769 < m~ < 0.7784...

ounds somewhat,

i.7769 < x~ < 0.7785

iracy is insuAicient, we could once again>d to the new bounds of the root ~x2.quire much more complicated calculations.>proximate solution of equations are more

' the following three. systems of numbers:iumbers, the set of all real numbers, andnumbers. Without leaving their respective

I

the error will not excethese bounds.

If the resulting acciapply the above meth<However, this would re

Other methods of al

already encountered abterms. To do so we mconcepts of algebra.

Let us first considerthe set of all rational tthe set of all complex

c=a+b

d =ah

Ill

a+x=b

the number 2 cannot bewell as from the systetn

:tion is not always possible.arming algebraic operationsdtiplication of polynomials,d in physics. Incidentally,d to consider addition and

driven, consisting either ofme arbitrary objects whichI'he operations of additionfor each pair of elementsent c from P is indicated,

n P. called their product:

n and multiplication defined:ions possess the following

e, i. e. for any a and b

=ha

i. e. for any a, b and c

Ik IIII III Ii. e. for any a and b

I'or any a and b, providedif the equation

is not always possible for example, 'divided by 5 without a remainder!, asof all positive real numbers, where subtra

The reader is already familiar with perfinot on nuinbers such as addition and mtand also addition of forces encounterein defining complex numbers we also hamultiplication of points of the plane.

In general terms, let a set P be lnumbers, of geometrical objects, or of sowe shall call the eiements of the set P. 'and multiplication are defined in P ifa, b from P one precisely defined clemand called their sum:

and a precisely defined element d froi

The set P with the operations of additiowithin it is called a field, if these opera five properties:

I. Both operations are commutativ

a+b=b+a, ab

II. Both operations are associative,

IV. Subtraction can be carried outa unique root of the equation

can be found in P.V Division can be carried out, i. e. I

a does not equal zero, a unique root c

Condition V mentions zero. IConditions I-IV. Indeed, if a isbecause of Condition IV a defisatisfies the equation

a+»

a itself is taken for b!. Since thichoice of the element a, we desi~

s element may depend upon theenate it by 0� i. e.

a+0,

If b is any other element of P, tunique element Ob for which

hen again there exists one such

,=b b+0�!

If we prove that 0, = Ob for arin the set P of an element, whicthe elements a at the saine tin

Let c be the root of the equi

a+x

which exists because of Conditioix IV; hence,

a+c

�TIt illllll TIIII !III Ill=bI b+0,

Comparing this to equation �! anIV there exists only one solutiorlinally reach the equality

d remembering that according to< of the equation b + x = b, we

Sl,",hWÃ~'~WIPbWSPl~~N'i...«I I I I I I 15 «« " " " " " " " " ii " " " " "ii 'i " " " " " ":i':i':i':i':i':i'«" "«":i' "«" «": " i' "." i' i« " iwiii i' iNi«" i'«" i' ij!i«

ts existence can be derived froman arbitrary element of P, thennite element exists in P which

iy a and b, then the existenceh plays the role of zero for allie, will be immediately proved.ition

Tl!

This ni. e. such

with arbitinstance, znumberspolynomiawe still wianother pi

Now let

f x!

if an on

f ! 0 ! =e ! q !Then,

ow proves that in any field P there is a zero element,an element 0 that for all a in P the equality

a+0=a

therefore Condition V becomes completely meaningful.:ady have three examples of fields � the field of rationalthat of real numbers, and that of complex numbers,�sets of all integers and of positive real numbers do notfields. Besides these three, an infinite number of other. For instance, many different fields are contained withinaf real numbers and of complex numbers; these are theiumerical tlelds. In addition some fields are larger thannplex numbers. The elements of these fields are no longerhers, but the fields I'ormed by them are used in mathematicalSere is one example of such a field.consider all possible polynomials

f x! = aox" + a,x" ' + + a�,x + a�

rary coinplex coefticients and of arbitrary degrees; for:ero-degree polynonuals will be represented by complexthemselves. Even if we add, subtract and multiplyls with complex coefficients by the rules we already know,ll not obtain a field, since division of a polynomial byolynomial with no remainder is not always possible.us consider ratios of polynomials

f x! u x! f x!u x! + g x!u{x!+

il x! u x! 0 x! v x!

holds andWe air

numbers,while the,constitutefields existthe fields iso-called ithat of corcalled numresearch. I

Let us

:tions whose numerator is equal

x!

ype are equal to one another.

s not equal zero, i. e. u x! g 0,

f x! v x!g x! « x!

ie above operations with rational:nts of the definition of a field;' rational functions with complexnumbers is totally contained inhose numerator and denominatoriply a complex number, and any9 in this form.i< field is either contained in theains it within itself: some of theinite number of elements.have to consider equations withinevitably the existence of rootsi,m. Thus, in some problems ofis with c ellicien

The role of zero is played by fra<to ~ero, i. e. fractions of the type

0

a

Obviously, all fractions of this <

u x!Finally, if a fraction doeU x

thenf x! u x!g x! u x!

It can easily be checked that ttfunctions satisfy all the requiremiso that we can speak of a /iield o~coefficients. The field of complexthis field, since a rational function ware zero-degree polynomials is sincomplex number can be presente<

One should not think that an~field of complex numbers or conlde'erent fields consist only of a l

Whenever fields are used, wecoefficients from these fields, andof such equations poses a probe

+ a�,x+ a�= 0

i coefficients from this field. Itnnot have more than n rootsither greater field. At the samed to a field Q in which our>f which may be multiple!. Even

...,,,I...<...,,i,,',,j,,l...

aox" + aix" +

be an equation of degree n witlturns out that this equation caeither in the field P or in any <:time the field P can be enlargeequation will have n roots some <

w' regi,�hol<js

This field P is called algebraically ciosed. The basic theoremof higher algebra shows that the field of complex numbers belongsto the set of algebraically closed fields.

S. Conclusion

Throughout this booklet we always discussed equations of acertain degree with one variable. The study of first-degree equationsis followed by that of quadratic equations in elementary algebra.In addition elementary algebra proceeds from a study of onefirst-degree equation with one variable to a system of two first-degreeequations with two variables and a system of three equations withthree variables. A university course in higher algebra continuesthese trends and teaches the methods for solving any system ofn first-degree equations with n variables, and also the methods ofsolving such systems of first-degree equations in which the nuinberof equations is not equal to the number of variables. The theoryof systems of lirst-degree equations and some related theoriesincluding the theory of matrices, constitute one special branch ofalgebra, viz. !ines algebra, which is widely used in geometryand other areas of mathematics, as well as in physics and theoreticalmechanics.

It must be remembered, that at present both the theory of

is a set with operations of addition and multiplication, in whichConditions I-IV from the definition of a field are valid; the setof all integers may be cited as an example. We already mentionedanother very significant branch of algebra, the group theory. A groupis a set with one algebraic operation, multiplication, which mustbe associative; division must be carried out without restrictions.

We often encounter noncommutarii:e algebraic operations, i. e. ones

ematic presentation of the fundamentals of the theory of.c equations and of linear algebra can be found in textbookster algebra. The following textbooks are most frequentlyiended:i. Kurosh, Higher Algebra, Mir Publishers, 1975 in English!.t'a. Okunev, Higher Algebra, "Prosveshchenie", 1966 in!.elementary presentation of the simplest properties of ringsds, mostly numerical, can be found in;Proskuryakov, Numbers and Polynotniais, "Prosveshchenie",

i Russian!.acquaintance with group theory may begin with:, Aleksandrov, Introduction to Group Theory, "Uchpedgiz".i Russian!.

Systalgebraion higlrecomrr

A,C1

RussianAn

and fielI. V.

1965 irAn,P S

1951 ir.

1U 1Ht: Kt ADt:K

Mir Publishers would be grateful for your comments on the content,tnslation and design of this book. We would also be pleased to:eive any other suggestions you may wish to make.

Our addres's is:gSR, 129820, Moscow 1-110, GSPrvy Rizhsky Pereulok, 2ir Publishers

trtre

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