Algebraic Equations of Arbitrary Degrees [A. G. Kurosh]

A.I Kypour AJII ESPAMtIECKHE IIPON3BOJlbHbIX H3gRTe.1bCTBO «Hayrca» YPABHEHNSI CTEl1EHEA A.U. Kurosh ALGE B RAIC EQUATIONS OF ARBITRARY DEGREES Translated from by V. Kisin the Russian t erst punhsned I/li Revised from the l975 Russian edition HQ OHZABQCKO.u A3slKC ! English translation, Mir Publishers, 1977 Evolution. Quadratic Equations te 2. Cubic Equations 19 3. Solution of Equations in Terms of Radicals and the Existence of Roots of Equations 22 The Number of Real Roots 24 Approximate Solution of Equations 27 6. Conclusion 35 Bibliography 36 .t 'ontents Preface 7 troduction 9 Complex Numbers 10 In 1. Fields 30 7. IIUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU . No proofs are have required copying almost . . this booklet does not m: popular mathematics book calls fc thorough consideration of all the del of calculations in all the exatnples described to his own examples. Even a ir the reader's concentration. check . Despite such an eke for light reading. initions and statements.Prefact the author's lecture to high Mathematics Olympiad at a review of the results and algebraic equations with due >f its readers. etc. application of the methods This booklet is a revision of school students taking part in thi Moscow State University.her algebra. It gives methods of the general theory of regard for the level of knowledge i included in the text since this would half of a university textbook on hi~ approach. IIUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUURRRRRRRRRRRR . school. a pupil knows the formula l'or solving quadratic <ations eqt ax'+ bx+ c= 0 ere a/0: wh 5+ x = rb' 4ac- 2a III i< l tations. and. Any pupil can solve jirst degree equations: if an equation ax+ are unl b =0 ~Jven. after factoring out x.Introduction A secondary school course of algebra is diversified but equations its focus. then its single root is X= is a Furthermore. Let us restrict ourselves to equations with one tnown.in which a 4 0. the third-degree cubic! equation ax + bx + ex=0 ich has one root x = 0. and recall what is taught in secondary. For example. is transformed a a quadratic equation ax +bx+c=O n wh inti . When complex numbers ced.legree quartic! equation ay +by A fourth-d +c=0 atic.i. where n is a positive integer.. modern science sees nothing mysterious cc' x2+ e r. numbers and were introdu existence..... students ofte. has required levelop and now constitutes one of the main parts bra taught at universities and pedagogical institutes.grai positive numbers in elementary arithmetic. even mathematicians doubted their actual >ce the term "imaginary numbers" which still vever.However. higher degree t mechanics ani arbitrary degr centuries to c of higher alger 1. However.. he> survives. He .~umber'. Is there a way to expand the tbers so that these equations also possess roots? !dy of mathematics at school the student sees numbers at his disposal constantly extended.1.. a doubt the justification for introducing these their actual existence... Hoi 1 =0 t of such equations.a. gati... we encounter and all the rl dgebraic equations in different branches of engineering.. may also be reduced to a quadratic equation = x.... 9 physics.ve. . Secondary school algebra lods of solving arbitrary equations of these degrees..... Very . called biquadr by setting y~ equation and Let us em types of cubi gives no met? Itore so of higher degrees. J is the simples realm In of nun his st< the system of starts with inn c l. Complex Numbers ' of algebraic equations is essentially based on the mplex numbers taught at high school. phasize once again that these are only very special c and quartic equations.ne... The theory theory of co: s. The theory of algebraic equations of an ee n. calculating the roots of the resulting quadratic then extracting their square roots... and if a chosen Fig. i. straight line. if the origin 0 is marked. e. e. We shall now proce outline of this last step. 1!. ' ~I i ibers in such a st as natural a not constructed obers. and the has a root equation after the introduction of negative nu x~ 2=0 m the way to ed to a general ted on a given unit of scale is can be put in has a root only after irrational numbers are adde All this completely justifies one more step enlarge the store of numbers. Thus. We must also define hov to carry out .previously had no roots. the equation =0 acquires a root only after fractions are introduce x+1=0 mbers. with a rea expresses in the chosen units of scale the dista and 0 if A lies to the right of the point 0. i. It is known that if a positive direction is fi. which this new But is it possible to expand the store of nun way that new numbers can be represented in ju manner by the points of a plane? So far we have a system of numbers wider than that of real nur We shall start by indicating the "material" with hat objects will system of numbers is to be "constructed". then each point A on this line 01 iIG l number which nce between A or the distance correspondence with its coordinate. the equation 2x 1 d. w act as new numbers. Just as the position ol defined by a single rea I' a point on a straight line is completely t number. a. b! of positive real numbers we can ly defined point Conversely. i. the new numbers. e. its coordinate. To seriously conyider we must merely define how to carry out i them. To do this let intersecting on the plane and set olf a unit of scale Fig. the position a plane can be defined by a pair of real us take two perpendicular straight lines. at the point 0 and on each of them of an arbitrary point on numbers. 2!. g.for each pair indicate a single in the first quadrant U U U UI I U U precise' . 2 II II I] two positive real numbers. e. Let fix the positive direction FIG. which one is to be their algebraic operations witt sum of two given points product. t e the selected units is completely defined by number a which gives in from this point to the c of scale the distance Irdinate axis the abscissaof point A!. etc. the s the plane themselves as these points as nuinbers. which point is to be the of the plane.implest way is to consider the points of points of the plane. sa axis the ordinate of the point A!. gives in the selected units of scale its and the number b which distance from the abscis. This enables us to talk further not < point A. At first these definitions may seem extrer artificial. b! correspondingto sev distinct points on the plane. e. b!. l. i. b! + c. Note that points on the abscissa are given by coordinates of the type a. d! be given on the plane.associative i. in order avoid the same pair of coordinates a. let us call the product of the given pi the point with the abscissa ac bd and the ordinate ad + bc. the brackets can be removed!. b! and c. d! = a + c. a. b + d! >ints e. e. b!.to eral S lrl ants axis the d b plane. Let us call their sum the point with the aha a+ c and the ordinate b+ d. we assume the abscissas of point quadrants 11 and 111 and the ordinates of points in quadr III and IV to be negative. Let us now define addition and multiplication of the pc on the plane. the sum and the produc three points are independentof the position of the brackets! distributive i. only such definitions will make it possibh realize our goal of taking square roots of negative real nuinl Let the points a. 0!. now we did not know how to define the sum and the proi of these points. Now we can also perform the operations of subtraction nn . t of and law :s it the and Dn the other hand. where a an are certain by if a oint iints nely to !ef s. lntil duct :issa real numbers. b!. e. and those on ordinate axis by coordinates of the type . Note that the of association for addition and multiplication of points makt possible to introduce in an unambiguous way the sum and product of any finite number of points on the plane. We are now able to define all the points on the plane pairs of real numbers. b!. be interchanged!. and the pairs of coordinates a. However. but simply of a p a. e. given by the coordinates a. fie abscissa axis. 0! b. 0!. we obtain. 0! = a . 6!. a. Thus. b a. e. plication of the 1! and let us find its square in the senseof multi points on the plane: . 6! c. let us consider which lies at a distance1 upwardof the point C this point by the letter i: i= . addition and multiplication of these points rei and multiplication of their abscissas. d! = a c. 0! a.The same is vali and division: a. i axis re resents If we assume that each oint 0! a0 6. defined above! The reader will easily see that the product as equality by the is even simpler of the point on the right-hand side of the last appoint c. e.d! ac+ bd c+d' d! bc ad! c+d J . 6! c. 0! + . 0! = a + 6. to the points of the type a.6!. By applying our definitions to the points on t.d! is indeed equal to thepointa. Let us denote with real numbers. 0! duce to addition d for subtraction i.It ht-hand side of to verify that the sum of the point on the rig d equal to the the first equality and the point c. 0! b. 1! i. 0! = ab. For example. 0! 6' of the abscissr in IIIII~ 'll I II IIIIIIIIIlIIIIIIIJIIII the point . d! is indee point a. a.by the divisor yields the dividend!. 0!+ . tructed is more :d the system of ith the operations s not difficult to by real numbers For example. and . b! = a. the fact that it is usually referred to as an "I does not in the least prevent it from actually exist The system of numbers we have just cons extensive than that of real numbers and is call complex ppmbers.axis. therefore a real. the point 1. b!. By virtue of the definition following is valid: a. 0!. e. the second The addend a. 0! lies on the abscissa ordinate axis. the i= . Thepoints ontheplane togethe w we have defined are called complex numbers. numbers we can >ns with complex b! 'I b! = b. Another value of tl nt . we can now 1!. By virtue of the definition of rnultiplic. 1!. It i prove that any complex number can be expressed and the nuinber i by means of these operations: us take point a. addend can be written in the form . and thus re'presents a real nuinber i~ = 1 >is root is given Hence. However. find the square root of 1.numbers: a + bi! + c + di! = a + c! + b + 4 c!i a+ bi! c+ di! = ac bd! + ad + $ a bi c+ i'I = c!+ . i. i. let of addition. 1 <]. Note that the poi we denoted as i. which i the plane. not on the 1. by the point imbers a number e. 0! lies on the abscissaaxis and is number a. is a precisely defined point or. we have found in our new system of m whose square is equal to a real number 1. i 1 1] I ~ ll I! ~ ~ By means of this andard notation of complex immediately rewrite t e above formulas for operatii '. ation.maginary unity" ing on the plane. n of multiplication nent with the law :cond of the above rule of binomial It should be noted that the above definitio of the points of the plane is in perfect agreer of distribution: if on the left-hand side of the @ w of distribution!. and then apply the equality i~ = de of we shall arrive precisely at the right-hand equation. a > 0. then the positive value of one :gative one to the to the positive value of the' other. Of course. this equation complex numbers. Returning to the solution of the quadratic i coefficients. always obtaming two distinct values. and the ne ~ to the following radical . Evolution. but of number. then V>are root of the :quation with real 0.Each of these two radicals pc which are combined with each other accordin rule: if b > 0. the reader will see that for. i.we can now say that where b2 4ac also has two distinct roots. we roots not only of the number l. the second I and red sI 2. Quadratic Equations can extract square any negative real ' a is a negative Having complex numbers at our disposal. b is taken in both any a and b both :oefficient in i will possesses two values =+v wherePa is the positivevalueof the sq positive number a. e. the first term on the right-hand side and the c is added be real numbers. Now we are able to take square roots of any rim '~immiIia s nlI wherethe positivevalueof the radical Pa + terms. this time complex. equations we calculate the product by the multiplication which itself stems from the la luce similar terms. II real number. A root of order n distinct y applicable to real numbers. e. In itn ortant >f any complex number has exactly n < n. it was proved that be:derived. 2 the formula for taking the square root + bi. f one us now cont lex values. for n o 2 bi in tertns of real values the nth root of a compie iliary real numbers. each other words. It can :omplex number m there exist exactly n > such that raised to the power n i. the following I extremely It can easily be checke satisfies the equation. one or no real numb a and parity of the inde> has three values: l and 22 i Thus. Roots of order n of cotnplex of radicals of certain aux no such formula can ever x number a+ . which are tplex numbers: the nth root of a real distinct values which in a general case that among these values 'there will be ers. This t eorem is equaH a particular case of con number a has precisely n are complex.ar equation are the numbers : 3+i. This formula reduces calculation quare roots of two positive real numbers. We know two. no formula exists which would express 1 2 c + In section 1 we gave of a complex number a of the root to extr'actinge @Unfortunately. the cube root y~ 1.et to theorem holds. the roots of oi xg = numbers indeed the problem of extracting roots of an index n from complex numbers. xg 1 2i d that each of these Therefore. n factors equal to this number!. I. depending on the sign of the number turn arbitrary positive integral be proved that for any < distinct complex number< if we take a product of yields the number a. ax The formula + bx + c = 0 x iis equation. Substituting this expression of x into cubic equation with respect to y. For th equations. expresses wit in terms of coefficients. where y is a new unknowi our equation. respectively. these values cani . which.Cubic Equations g3. ig quadratic equations is also valid for iird-degree equations. . We transform tl a x=y 3 a. although more h radicals the roots of these equations 'his formula is also valid for equations for solviti complex coefficients. 'I with arbitrary complex co Let an equation efficients. setting be given. since the coeffi of the lirst power of y ani the numbers tb> written '+ il= 2a 27 ab 3 +c i. e. the equation can be as + =0 + p + ' 27+ the three cube radicals has three values. The coefficient l the absolute term will be. usually called cubic erive a formula. we obtain a is simpler. we can also d complicated. We know that each of However. iot be combined in an arbitrary manner. which cient of y~ will be zero. then the expression has three distinct rea + 4 27 :e the expression is under the square root extracting this root will yield a complex !f the two cube root signs. The practical sigr small. It the equation y +py+ . Sini sign in the formula. i. can be shown that if a=0 I roots. constitute P 3 The together to obtain a three roots of our eq numerical coefficients complex. e. each cubic equation with has three roots. Indeed.se two values of the radicals the number must be added root of the equation. Example. We mentioned of cube roots of complex numbers requires i. obviously sc a multiple root. the result is neg: formula yields in the 3 3 P 27 I' IINSIS . The equ ation x itive. Therefore. 19x+ 30 = 0 The first of the cube radicals i. will be negative. Thus we obtain the uation. which in a general case are mmeof these roots may coincide. but this can be done only approxiinately. let th< iificance of the above formula is extremely : coefficients p and q be real numbers. number under each < above that extraction trigonometric notatioi by means of tables. e. In practice the above formula for solving cubic equationsyields the roots of equations only when the expression + 4 27 is po- sitive or equal to zero.this formula. we obtain += 4 27 P 196 = 14' and therefore 3 IIIIIIIIIIIIIEIIIITlfffmff ItffffffftffI value of the second radical will be the number the roots of the reduced equation is yi =3+ 1. In the first instance the equation has one real and two complex roots. But a direct verification are the integers 2. For instance. 3 and demonstrates that these roots S. Kxatnple. we . we can obtain the other two in many different ways. in the second instance aH the roots are real but one of thetn is multiple. We want to solve the cubic equation x' 9x' + 36x 80 =0 Setting x=y+3 we obtain the "reduced" equation ys+ 9y 26 =0 Applying the formula. and one of 1! =2 Knowing one of the roots of the cubic equation. x" '+ a.ing values of the radicals. Or we may divide the left-hand side of the reduced equation by y 2.valuesof the secondradical and add up the mutually correspond. wheren is a positive integer. yielding only approximate values of the roots of an equation.2+ i+12and2 i~12 Of course. much more often they have to be calculated approximately. calculation of radicals is not always as easy as in the carefully selected example discussed above. Involving still more "multi-storied" radicals. this formula I] ] l ] ]i ] '] I] ] '1i] ] ' I i']t ll higherdegrees. when the followin s ectacular result was roved: . Solution of Equations in Terms of Radicals and the Existence of Roots of Equations Quartic equations also allow a formula to be worked out which expresses the roots of these equations in terms of their coefficients. after which we only need to solve a quadratic equation. Either of these methods will demonstrate that the other two roots of our reduced equation are the numbers 1+ <+12and 1 i+12 Therefore. is aox" + a. 4.x+ a=O The searchcontinued unsuccessfully until the beginningof the 19th century.x" + + a. the roots of the original cubic equation are the numbers 5. Note that a general form of an equation of degree n. Voronoi Russ !-1908!.is i greater than that of the numbers which can be written in s of radicals. i.and evenfor cubic equations . G. inade valuable contribu- in this field. e radicands involve only integral or fractional numbers. and radi Ther form are are i mucl term 'I ian mathematiciansE. Ii 'he theory of algebraic numbersis an important branch of algebra. the store of numbers. I. e.s far as practical determination of roots of equations is erned. G. however complicated. this equation is "unsolvable in terms of radicals-. Zolotarev 847-1878!.n addition. Such ie equation x' 4x 2 I equa who if th is tF =0 t can be proved that this equation has five roots. but.F. unsolvable in terms of radicals. se roots are not expressible in radicals. an ition can be written of degree n with integral coefficients. Numerous methods of oximatesolution of equationssuffice. three real two coinplex. tions . pion~ . Chebotarev 894-1947! 86l efore.the absenceof formulas for solving nth degreeequations e n! 5 causes no serious difficulties. both real and complex. for any n. but none of these roots can be expressed in :als.bel 802-1829! proved that deriving general formulas for ng equations of degrees n o 5 in terms of radicals was solvi issible. methods are much auicker than th li 'o ornn%q ~ ~~ i conc wher appr these . greater than or equal to five. Galois 811-1832! demonstrated the existence of impc tions with integral coefficients.. N. ilso found the conditions under which the equation can be equa He shall only mention that presently Soviet mathematicians are :ering in the development of group theory. which the roots of equationswith integral coefficients such numbers called algebraic as opposed to transcendent numbers which not the roots of any equations with integral coefficients!. .x" + a. at pre there exist several dozen dilferent proofs of this theorem. but it provides no practical method solving the roots of equations.x" '+ side oots + a= ao x m..Ja. will be the r The Number of Real Roots The basic theorem of higher algebra has important applicat in theoretical research. The following theorem h< Any equation of degreen with any numerical coefficients has n r complex or. although these proofs were perfe to complete rigorousness only in the 19th century. mentioned in the basic theo means the following. form multiple roots. This theorem is called the basic theorem of higher alg~ It was proved by D'Alembert 717-1783! and Gauss 777-1 as early as the 18th century.! x mq! .. + a. The concept of a multiple root. which possessed no real or complex roots. However. some of these roots may coin i. x 1 Conversely. 855! cted . |x.x the numbers is given for the left-hand t«c« . However. ions for ems :nts.sent rem. if such a factorization of our e uation /5. real. the s of real numbers would have to be extended. +a1x+ a=0 .real tore s 1s any !lds: oots> cide. then the left-hand side of the equ.bra. thi unnecessary since cotnplex numbers are sufhcient to solve equation with numerical coefficients.. It can be proved that if an nth de equation aox" + a1x" '+ ition has n roots txnz. . gree If there were equations with numerical coefficients... in certain cases. many technical prob] require information about the roots of equations with real coeffich . either or complex. e.. can be factored in the following manner: . f! =2 5 2 + 2. e. how many and approximately located? We can answer these questions as follows. x! = aox" + a. :ly. into the expressionfor f x!. 2+ 1 = 7 : a graph of the polynomial f x!. if f x! = x' Sx' + 2x + 1 and m=2. To do this we ordinateaxes on the plane see above! and. having i value the Ix and calculated s I I I I I]I I a corres II I I I I I I I » ]IqlI I I I I i ondin value m Let us plol choose the coc selected for x I ]II I I I I I question. i. We already know that it has n roots. f u!! for all of them satisfied with a finite number of points. after performing all the substituting it i operations.x" ++ a.degree equation be given uox" + a>x" ' + +a Let an nth ix + a= 0 efficients.x+ a iiliar with the concept of function will understand he left-hand side of the equation as a function of Taking for x an arbitrary numerical value m and The reader fan that we treat t the variable x. mark on the plane the points to them and then draw through them as smooth Unfortunat» of »x one cann» and must be: sake of simplic integral values corresponding . m real roots? If so. For the ity we can first select several positive and negative of »x in succession. we of the polynon arrive at a certain iial f x! number which is called the value and is denoted as f cx!. Thus. since there are an infinite number of the values >t hope to lind the points n. he polynoinial on the left-hand side of our equation having real cc Are any of the where are they Let us denote t by f x!. Plot a graph of the polynomial f x! = x 5x m 2x + 1 Here ao! = 1. Let us compile a table of values of the polynomial f x! and plot a graph Fig. e. which fall between 1 and 5. we could overlook several additional . The graph demonstrates that all the three roots a.. A = 5. then +1 However. a2. Example. 3!. for this particular example we can restrict ourselves to only those values of m. B= A I ~0I acr. and thus it was sufficient just to look ht the table of values of f a!. and a3 i! li 1I i! 1 I! 1 I neighbouringvaluesof a for which the numbersf m! have opposite signs. and thus B = 6. Actually.. we might think that owing to the imperfection of our graph we traced the curve knowing only seven of its points!.and [a~ = a for a < 0! and A is the greatest of the absolute values of all the other coefficients a.it is often apparent that these bounds are too wide. If in our example we found less than three points of intersection of the graph with the abscissaaxis. it also has at least one negative root. Thus. Any equation of an odd degree with real coefftcients has at least one real root.cated between any given equation and even the number of roots lo thods will numbers here. the equation x' 8x +x 2=0 has at least one positive root. if our equation is of an even degree. These me Sometimes the following theorems are useful since they give some information on the existence of real and even positive roots. In addition. not be stated a and b. For < If1 4 . coefftcient ao and the absolute term a in an equation with real coefftcients have opposite signs.the equation -x +2x' se neighbouring integers . where a b. the equation has at least one positive root. If the leading. while .tion x 1=0 In the previous section we found tho. between which the real roots of the equa I0 e roots of this equation :xample. The 5x same method +2x+ allows I= th to be found with greater accuracy. let us take the +7x x are located. 9.293.8.sive values >scissa axis.1.8 n is given. than the ones we aln 5x + 2x + 1 obtain = 0. This has led to the development of various . this approac cumbersome calculations which soon becoine practica ageable. '0. For this purpose we shall calculate our r accuracy of one-tenth. < 0. h involves lly unman- for this root narrower f x! = x for x = 0. However. bounds oot to the lues of the 0 < m. Therefore . The method is as follows. we can find between which two of these succes of x the graph of the polynomial f x! intersectsthe af methods of :ions much :thods and of the Ful to find i. and the values of f i are given above. tl to any accuracy we want. But first it is use >ady know. calculating approximate values of real roots of equai quicker. we can find the value of the the accuracy of one-hundredth. f one-tenth.7 < >x. root u2 to 0. f.f b! i! and f b! In this case a =0.9.088 and therefore.since the signs of these values of f x! a: 0.8! = 0. b = 0. e. An equation of degree The bound c is calculated by means of the formul. one-thousandth or. he will f.7! re difFerent. bf a! af ! f a! . If the reader calculates the val polynomial ieoretically.7.2. we can now calculate the root u2 to the accuracy o Proceeding further. Below we present the simplest of these mi immediately apply it to the calculation of the root cubic equation considered above. < 1. 0. moreover. disappears. the secondderivativef" x! will usually have for x = a and x = b. ynomial f ' x!. h is called the voted as f" x!. while the exponent itself is reduced by u: the absolute term a. however if the . i. as we know.the secondformula.x" + 2a-zx+ a derivative of this polynomial. the one same. If the bounds a.llJill lllTIIB I IIllI chosen. +i 1! + [n 21 azx' a -z and denote it polynomial is derived from j x! by the following r ax" ' of the polynomial f x! is multiplied by n k of x. d must signs of f" b! in which the bound a is bound andf b! coincide. while the signs of f a! am different. that a= axo. e. b are chosen sufftciently I f b! will be 1 f a! are the other. involving the the same sign b. whic secondderivative of the polynomial f x! and is de> Thus.iction of a new The formula for the bound d requires the introdv :re. If the signs of f" a! am i. since we can consider We can again take the derivative of the pol This will be a polynomial of degree n 2!. This ule: each term the exponent nity.x" '+ azx" + + an azx Let us call the polynomial of degree n f'4! as f'{x!. for the above polynomial f x! = x' 5x obtain x! 3xz 10x + 2 f" x! = 6x 10 close to each lIIIIJlllRllllll. in essence concept which will play only an auxiliary role h< lied differential it belongs to a dillerent branch of mathematics cai calculus. must be calculated with the first formula used. +2x+ 1 we = nnvx" ' + in 1! a. Let a polynomial of degree n be given zn-sx+ f x! = aox"+a. ounds somewhat. To do so we m concepts of algebra..088= 0.7784.. >proximate solution of equations are more If the resulting acci apply the above meth< III ' III IIII III its of algebraic equations. e calculated u= It follows. Thus narrower before: for the root than those wi < m~ < 0. and numbers. systems of numbers: iumbers. which we have ove.. th.0215. x2 we have found the following : knew 7769 d =0. if we widen these b i.7785 it if we take for ei~ the arithmetic mean.8! = < be l. therefore.7777 ed 0.. the set of all real numbers..8 bounds.0008. Sincef'. equal to half the difference of the error will these bounds. i. quire much more complicated calculations. can be considered in more general ust introduce one of the most important However. of th bounds.7784 . or.80.= 0.. Without leaving their respective I Let us first consider the set of all rational t the set of all complex h . we obtain 0. iracy is insuAicient. ' the following three.second formula for the bound d must and f b! negative. half the sum. the ~ used. 0. we could once again >d to the new bounds of the root ~x2. e.08.7769 < x~ < 0. not exce this would re Other methods of al IIII The problem of roc already encountered ab terms. b and c Ik IIII i. e. i. Both operations are associative. arming algebraic operations dtiplication of polynomials. Subtraction can Ill be carried out a unique root of the equation a+x=b I'or any a and b. for any a and b ab III I IV. of all positive real numbers. i. e. ' divided by 5 without a remainder!. ' and multiplication are defined in P if a.the number 2 cannot is not always possible for example. for any a. In general terms. e. for any a and I. I . d to consider addition in defining complex numbers we also ha multiplication of points of the plane. where subtra The reader is already familiar with perfi not on nuinbers such as addition and mt and also addition of forces encountere d in physics. a unique root c out. Incidentally. II. called their product: d =ah The set P with the operations of additio within it is called a field. we shall call the eiements of the set P. let a set P be l numbers. of geometrical objects. or of so and driven. if these opera five properties: n and multiplication defined :ions possessthe following e. e. provided can be found in P. Both operations are commutativ b =ha a+b=b+a. b from P one precisely defined clem and called their sum: c=a+b and a precisely defined element d froi n P. V Division can be carried if the equation a does not equal zero. consisting either of me arbitrary objects which I'he operations of addition for each pair of elements ent c from P is indicated. i. as be well as from the systetn :tion is not always possible. ".=b If b is any other element of P. whic the elements a at the saine tin Let c be the root of the equi ition a+x x IV. Indeed.hWÃ~'~WIPbWSPl~~ N'i. which hence. " "" "" "" "ii" "" ""ii'i" "" "" ":i':i':i':i':i':i'«" "«":i' "«" «":" i'".. we «II I II I 15 «« Comparing this to equation ! an IV there exists only one solutior linally reach the equality Sl. a itself is taken for b!. Since thi choice of the element a."i'i«" iwiii i'iNi«" i'«" i'ij!i « . = Ob for ar in the set P of an element. d remembering that according to < of the equation b + x = b. e. then the existence h plays the role of zero for all ie. hen again there exists one such . I an arbitrary element of P. we desi~ a+0. exists because of Conditioi a+c TIt illllllTIIII!IIIIll Tl! I =b b+0. if a is nite because of Condition element exists in P which IV a defi satisfies the equation a+» s element may depend upon the enate it by 0 i. If we prove that 0.. will be immediately proved. t unique element Ob for which b+0 ! iy a and b. then Conditions I-IV.ts existence can be derived from Condition V mentions zero. x +a rary coinplex coefticients and of arbitrary degrees. these are the iumerical tlelds.x" ' + + a. constitute fields exist the fields i so-called i that of cor called num research. e. consider all possible polynomials while the. numbers. ll not obtain a field. z numbers polynomia we still wi another pi Now let f x! if an f ! 0 ! =e ! q ! Then. an element 0 that for all a in P the equality This n i. and that of complex numbers.but the fields I'ormed by them are used in mathematical Sere is one example of such a field. In addition some fields are larger than nplex numbers. Even if we add. I Let us f x! = aox" + a. an infinite number of other . :ady have three examples of fields the field of rational holds and We air that of real numbers. subtract and multiply ls with complex coefficients by the rules we already know.ow proves that in any field P there is a zero element. The elements of these fields are no longer hers. for :ero-degree polynonuals will be represented by complex themselves. many different fields are contained within af real numbers and of complex numbers. For instance. Besides these three. us consider ratios of polynomials with arbit instance. since division of a polynomial by olynomial with no remainder is not always possible. sets of all integers and of positive real numbers do not fields. f x! il x! + u x! u x! f x!u x! + g x!u{x! 0 x!v x! on . such a+0=a therefore Condition V becomes completely meaningful. The field of complex hose numerator this field. e..hol<js . and any 9 in this form.. At the same d to a field Q in which our be an equation of degree n witl turns out that this equation ca either in the field P or in any < time the field P can be enlarge >f which may be multiple!. all fractions of this < s not equal zero. i... and denominator iply a complex number.. u x! g 0..l. we coefficients from these fields.j... since a rational :nts of the definition of a field..i. if a fraction u x! Ux doe then f x! v x! g x! « x! f x! g x! ie above operations with rational u x! u x! ' rational functions with complex numbers is totally contained in It can easily be checked that tt functions satisfy all the requiremi so that we can speak of a /iield o~ coefficients.. in some problems of function w are zero-degree polynomials is sin complex number can be presente One should not think that an~ field of complex numbers or conl de'erent fields consist only of a l Whenever fields are used.. have to consider equations with inevitably the existence of roots i. and of such equations poses a probe is with c ellicien .I. e.<. Finally.. + a. i< field is either contained in the ains it within itself: some of the inite number of elements. Thus.. Obviously. fractions of the type 0 x! a ype are equal to one another.x+ a= aox" + aix" 0 + ither greater field. Even equation will have n roots some < i coefficients from this nnot have more than field..'. It n roots w' regi.m..:tions whose numerator is equal The role of zero is played by fra< to ~ero. i.. in which Conditions I-IV from the definition of a field are valid. We often encounter noncommutarii:ealgebraic operations. A group is a set with one algebraic operation. which must be associative. It must be remembered. !ines algebra. The theory of systems of lirst-degree equations and some related theories including the theory of matrices. We already mentioned another very significant branch of algebra. ones . as well as in physics and theoretical mechanics. and also the methods of solving such systems of first-degree equations in which the nuinber of equations is not equal to the number of variables. The study of first-degreeequations is followed by that of quadratic equations in elementary algebra. division must be carried out without restrictions.This field P is called algebraically ciosed. that at present both the theory of is a set with operations of addition and multiplication. the set of all integers may be cited as an example. which is widely used in geometry and other areas of mathematics. A university course in higher algebra continues these trends and teaches the methods for solving any system of n first-degree equations with n variables. In addition elementary algebra proceeds from a study of one first-degreeequation with one variable to a systemof two first-degree equations with two variables and a system of three equations with three variables. Conclusion Throughout this booklet we always discussed equations of a certain degreewith one variable. The basic theorem of higher algebra shows that the field of complex numbers belongs to the set of algebraically closed fields. e. constitute one special branch of algebra. S. the group theory. multiplication. viz. i. Syst algebrai on higl recomrr A. Kurosh. V.c equations and of linear algebra can be found in textbooks ter algebra. Higher Algebra. i Russian!. P S 1951 ir. Introduction to Group Theory. i Russian!. Okunev.C 1 Russian An and fiel I. acquaintance with group theory may begin with: . 1966 in !. Numbers and Polynotniais. The following textbooks are most frequently iended: i. Proskuryakov. Aleksandrov. elementary presentation of the simplest properties of rings ds. can be found in. "Prosveshchenie". "Prosveshchenie".1975 in English!. mostly numerical.Mir Publishers. Higher Algebra. "Uchpedgiz". t'a. 1965 ir An.ematic presentation of the fundamentals of the theory of . . 2 ir Publishers trt re U. We would also be pleased to :eive any other suggestions you may wish to make. 129820. Pe M . GSP rvy Rizhsky Pereulok.1U 1Ht: Kt ADt:K Mir Publishers would be grateful for your comments on the content. tnslation and design of this book. Moscow 1-110. Our addres's is: gSR. employing the visual language graphs. Meaty. Graphs. Needs no more than "0" level mathematics on e part of the reader.Other Books by MIR PUBLISHERS from LITTLE . E. Useful for sixth-form reading. ar lh ol th . Solutions to Problems. Derivatives. Integrals. MATHEMATICS LIBRARY Series Shilov ALCULUS OF RATIONAL An introduction FUNCTIONS to the principal C G concepts of mathematical ialysis the derivative and the integral! within the comparatively nited Iield of rational functions. Contents. Contents. . and first-year students whose courses delude mathematics. Soiilnky 'HE METHOD OF MATHEMATICAL INDUCTION One of the popular lectures in inathematics. Proofs of Some Theorems in Elementary algebra by the Method of Mathematical Induction. S. Problems of an Arithmetical feature.the ideas of induction ave general educational value. Will interest sixth-formers oing "A" level mathematics.. widely used in chool extracurricular maths circles. and will interest a broad readership ot familiar with mathematics. Describes the method of mathematical induction o widely used in various fields of mathematics from elementary chool courses to the most complicated investigations. Problems on the 'roof of Inequalities. Apart from eing indispensiblefor study of mathematics.Trigonometric and Algebraic Problems. Proofs of Identities. Documents Similar To Algebraic Equations of Arbitrary Degrees [A. G. 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