The Development of Abstract Algebra

Karen Hunger Parshall

1 Introduction

What is algebra? To the high school student encoun- tering it for the first time, algebra is an unfamiliar abstract language ofx’s andy’s,a’s andb’s, together with rules for manipulating them. These letters, some of them variables and some constants, can be used for many purposes. For example, one can use them to express straight lines as equations of the form y = ax+b, which can be graphed and thereby visualized in the Cartesian plane. Furthermore, by manipulating and interpreting these equations, it is possible to determine such things as what a given line’s root is (if it has one)—

that is, where it crosses thex-axis—and what its slope is—that is, how steep or flat it appears in the plane relative to the axis system. There are also techniques for solving simultaneous equations, or equivalently for determining when and where two lines intersect (or demonstrating that they are parallel).

Just when there already seem to be a lot of tech- niques and abstract manipulations involved in deal- ing with lines, the ante is upped. More complicated curves like quadratics,y = ax²+bx+c, and even cubics,y = ax³+bx²+cx+d, and quartics,y = ax⁴+bx³+cx²+dx+e, enter the picture, but the same sort of notation and rules apply, and similar sorts of questions are asked. Where are the roots of a given curve? Given two curves, where do they intersect?

Suppose now that the same high school student, hav- ing mastered this sort of algebra, goes on to university and attends an algebra course there. Essentially gone are the by now familiar x’s,y’s,a’s, and b’s; essen- tially gone are the nice graphs that provide a way to picture what is going on. The university course reflects some brave new world in which the algebra has some- how become “modern.” Thismodernalgebra involves abstract structures—groups[I.3 §2.1],rings[III.81 §1], fields[I.3 §2.2], and other so-called objects—each one defined in terms of a relatively small number of axioms and built up of substructures like subgroups, ideals, and subfields. There is a lot of moving around between these objects, too, via maps like group homomor- phisms and ringautomorphisms[I.3 §4.1]. One objec- tive of this new type of algebra is to understand the underlying structure of the objects and, in doing so, to

build entire theories of groups or rings or fields. These abstract theories may then be applied in diverse set- tings where the basic axioms are satisfied but where it may not be at all apparent a priori that a group or a ring or a field may be lurking. This, in fact, is one of modern algebra’s great strengths: once we have proved a gen- eral fact about an algebraic structure, there is no need to prove that fact separately each time we come across an instance of that structure. This abstract approach allows us to recognize that contexts that may look quite different are in fact importantly similar.

How is it that two endeavors—the high school analy- sis of polynomial equations and the modern algebra of the research mathematician—so seemingly different in their objectives, in their tools, and in their philosoph- ical outlooks are both called “algebra”? Are they even related? In fact, they are, but the story ofhowthey are is long and complicated.

2 Algebra before There Was Algebra:

From Old Babylon to the Hellenistic Era Solutions of what would today be recognized as first- and second-degree polynomial equations may be found in Old Babylonian cuneiform texts that date to the sec- ond millenniumb.c.e. However, these problems were neither written in a notation that would be recogniz- able to our modern-day high school student nor solved using the kinds of general techniques so characteris- tic of the high school algebra classroom. Rather, par- ticular problems were posed, and particular solutions obtained, from a series of recipe-like steps. No general theoretical justification was given, and the problems were largely cast geometrically, in terms of measurable line segments and surfaces of particular areas. Con- sider, for example, this problem, translated and tran- scribed from a clay tablet held in the British Museum (catalogued as BM 13901, problem 1) that dates from between 1800 and 1600b.c.e.:

The surface of my confrontation I have accumulated:

45 is it. 1, the projection, you posit. The moiety of 1 you break, 30 and 30 you make hold. 15 to 45 you append: by 1, 1 is equalside. 30 which you have made hold in the inside you tear out: 30 the confrontation.

This may be translated into modern notation as the equationx²+1x= ³₄, where it is important to notice that the Babylonian number system is base 60, so 45 denotes ⁴⁵₆₀=³₄. The text then lays out the following algorithm for solving the problem: take 1, the coeffi- cient of the linear term, and halve it to get¹₂. Square¹₂

to get ¹₄. Add¹₄to³₄, the constant term, to get 1. This is the square of 1. Subtract from this the¹₂which you mul- tiplied by to get ¹₂, the side of the square. The modern reader can easily see that this algorithm is equivalent to what is now called the quadratic formula, but the Baby- lonian tablet presents it in the context of a particular problem and repeats it in the contexts of other partic- ular problems. There are no equations in the modern sense; the Babylonian writer is literally effecting a con- struction of plane figures. Similar problems and simi- lar algorithmic solutions can also be found in ancient Egyptian texts such as the Rhind papyrus, believed to have been copied in 1650b.c.e. from a text that was about a century and a half older.

There is a sharp contrast between the problem-ori- ented, untheoretical approach characteristic of texts from this early period and the axiomatic and deductive approach thateuclid[VI.2] introduced into mathemat- ics in around 300b.c.e.in his magisterial, geometrical treatise, theElements. (Seegeometry[II.2] for a fur- ther discussion of this work.) There, building on explicit definitions and a small number of axioms or self- evident truths, Euclid proceeded to deduce known—

and almost certainly some hitherto unknown—results within a strictly geometrical context. Geometry done in this axiomatic context defined Euclid’s standard of rigor. But what does this quintessentially geometrical text have to do with algebra? Consider the sixth propo- sition in Euclid’s Book II, ostensibly a book on plane figures, and in particular quadrilaterals:

If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.

While clearly a geometrical construction, it equally clearly describes two constructions—one a rectangle and one a square—that have equal areas. It therefore describes something that we should be able to write as an equation. Figure 1 gives the picture correspond- ing to Euclid’s construction: he proves that the area of rectangle ADMK equals the sum of rectangles CDML and HMFG. To do this, he adds the square on CB—

namely, square LHGE—to CDML and HMFG. This gives square CDFE. It is not hard to see that this is equiva- lent to the high school procedure of “completing the square” and to the algebraic equation(2a+b)b+a²= (a+b)², which we obtain by setting CB = a and

A C B D

E G F

H

L M

K

Figure 1 The sixth proposition from Euclid’s Book II.

BD =b. Equivalent, yes, but for Euclid this is a spe- cificgeometricalconstruction and a particulargeomet- rical equivalence. For this reason, he could not deal with anything but positive real quantities, since the sidesof a geometrical figure could only bemeasuredin those terms. Negative quantities did not and could not enter into Euclid’s fundamentally geometrical mathe- matical world. Nevertheless, in the historical literature, Euclid’s Book II has often been described as dealing with “geometrical algebra,” and, because of our easy translation of the book’s propositions into the lan- guage of algebra, it has been argued, albeit ahistori- cally, that Euclidhad algebra but simply presented it geometrically.

Although Euclid’s geometrical standard of rigor came to be regarded as a pinnacle of mathematical achieve- ment, it was in many ways not typical of the math- ematics of classical Greek antiquity, a mathematics that focused less on systematization and more on the clever and individualistic solution of particular prob- lems. There is perhaps no better exemplar of this than archimedes [VI.3], held by many to have been one of the three or four greatest mathematicians of all time. Still, Archimedes, like Euclid, posed and solved particular problems geometrically. As long as geom- etry defined the standard of rigor, not only negative numbers but also what we would recognize as poly- nomial equations of degree higher than three effec- tively fell outside the sphere of possible mathemati- cal discussion. (As in the example from Euclid above, quadratic polynomials result from the geometrical pro- cess of completing the square; cubics could conceiv- ably result from the geometrical process of completing the cube; but quartics and higher-degree polynomials could not be constructed in this way in familiar, three- dimensional space.) However, there was another math-

ematician of great importance to the present story, Diophantus of Alexandria (who was active in the mid- dle of the third centuryc.e.). Like Archimedes, he posed particular problems, but he solved them in an algorith- mic style much more reminiscent of the Old Babylo- nian texts than of Archimedes’ geometrical construc- tions, and as a result he was able to begin to exceed the bounds of geometry.

In his textArithmetica, Diophantus put forward gen- eral, indeterminate problems, which he then restricted by specifying that the solutions should have partic- ular forms, before providing specific solutions. He expressed these problems in a very different way from the purely rhetorical style that held sway for centuries after him. His notation was more algebraic and was ulti- mately to prove suggestive to sixteenth-century math- ematicians (see below). In particular, he used special abbreviations that allowed him to deal with the firstsix positive and negative powers of the unknown as well as with the unknown to the zeroth power. Thus, what- ever his mathematics was, it was not the “geometrical algebra” of Euclid and Archimedes.

Consider, for example, this problem from Book II of theArithmetica: “To find three numbers such that the square of any one of them minus the next fol- lowing gives a square.” In terms of modern notation, he began by restricting his attention to solutions of the form(x+1,2x+1,4x+1). It is easy to see that (x+1)²−(2x+1)=x²and(2x+1)²−(4x+1)=4x², so two of the conditions of the problem are immediately satisfied, but he needed(4x+1)²−(x+1)=16x²+7x to be a square as well. Arbitrarily setting 16x²+7x= 25x², Diophantus then determined thatx=⁷₉gave him what he needed, so a solution was¹⁶₉,²³₉,³⁷₉, and he was done. He provided no geometrical justification because in his view none was needed; asinglenumerical solu- tion was all he required. He did not set up what we would recognize as a more general set of equations and try to find all possible solutions.

Diophantus, who lived more than four centuries after Archimedes’ death, was doing neither geometry nor algebra in our modern sense, yet the kinds of problems and the sorts of solutions he obtained for them were very different from those found in the works of either Euclid or Archimedes. The extent to which Diophantus created a wholly new approach, rather than drawing on an Alexandrian tradition of what might be called “algo- rithmic algebraic,” as opposed to “geometric algebraic,”

scholarship is unknown. It is clear that by the time Dio- phantus’s ideas were introduced into the Latin West in

the sixteenth century, they suggested new possibilities to mathematicians long conditioned to the authority of geometry.

3 Algebra before There Was Algebra:

The Medieval Islamic World

The transmission of mathematical ideas was, however, a complex process. After the fall of the Roman Empire and the subsequent decline of learning in the West, both the Euclidean and the Diophantine traditions ulti- mately made their way into the medieval Islamic world.

There they were not only preserved—thanks to the active translation initiatives of Islamic scholars—but also studied and extended.

al-khw¯arizm¯ı [VI.5] was a scholar at the royally funded House of Wisdom in Baghdad. He linked the kinds of geometrical arguments Euclid had presented in Book II of hisElementswith the indigenous problem- solving algorithms that dated back to Old Babylonian times. In particular, he wrote a book on practical math- ematics, entitledal-Kit¯ab al-mukhtas.ar f¯ı h.is¯ab al-jabr wa’l-muq¯abala (“The compendious book on calcula- tion by completion and balancing”), beginning it with a theoretical discussion of what we would now recog- nize as polynomial equations of the first and second degrees. (The latinization of the word “al-jabr” or “com- pletion” in his title gave us our modern term “alge- bra.”) Because he employed neither negative numbers nor zero coefficients, al-Khw¯arizm¯ı provided a system- atization in terms of six separate kinds of examples where we would need just one, namelyax²+bx+c=0.

He considered, for example, the case when “a square and 10 roots are equal to 39 units,” and his algo- rithmic solution in terms of multiplications, additions, and subtractions was in precisely the same form as the above solution from tablet BM 13901. This, how- ever, was not enough for al-Khw¯arizm¯ı. “It is neces- sary,” he said, “that we should demonstrate geomet- rically the truth of the same problems which we have explained in numbers,” and he proceeded to do this by “completing the square” in geometrical terms rem- iniscent of, but not as formal as, those Euclid used in Book II. (Ab¯u K¯amil (ca. 850–930), an Egyptian Islamic mathematician of the generation after al-Khw¯arizm¯ı, introduced a higher level of Euclidean formality into the geometric–algorithmic setting.) This juxtaposition made explicit how the relationships between geomet- rical areas and lines could be interpreted in terms of numerical multiplications, additions, and subtractions,

a key step that would ultimately suggest a move away from the geometricalsolution of particular problems and toward analgebraic solution ofgeneral types of equations.

Another step along this path was taken by the math- ematician and poet Omar Khayyam (ca. 1050–1130) in a book he entitledAl-jabr after al-Khw¯arizm¯ı’s work.

Here he proceeded to systematize and solve what we would recognize, in the absence of both negative num- bers and zero coefficients, as the cases of the cubic equation. Following al-Khw¯arizm¯ı, Khayyam provided geometrical justifications, yet his work, even more than that of his predecessor, may be seen as closer to a general problem-solving technique for specific cases of equations, that is, closer to the notion of algebra.

The Persian mathematician al-Karaj¯ı (who flourished in the early eleventh century) also knew well and appreciated the geometrical tradition stemming from Euclid’s Elements. However, like Ab¯u-K¯amil, he was aware of the Diophantine tradition too, and synthe- sized in more general terms some of the procedures Diophantus had laid out in the context of specific exam- ples in theArithmetica. Although Diophantus’s ideas and style were known to these and other medieval Islamic mathematicians, they would remain unknown in the Latin West until their rediscovery and trans- lation in the sixteenth century. Equally unknown in the Latin West were the accomplishments of Indian mathematicians, who had succeeded in solving some quadratic equations algorithmically by the beginning of the eighth century and who, like Bragmagupta four hundred years later, had techniques for finding inte- ger solutions to particular examples of what are today called Pell’s equations, namely, equations of the form ax²+b=y², whereaandbare integers andais not a square.

4 Algebra before There Was Algebra: The Latin West

Concurrent with the rise of Islam in the East, the Latin West underwent a gradual cultural and polit- ical stabilization in the centuries following the fall of the Roman Empire. By the thirteenth century, this relative stability had resulted in the firm entrench- ment of the Catholic Church as well as the establish- ment both of universities and of an active economy.

Moreover, the Islamic conquest of most of the Iberian peninsula in the eighth century and the subsequent establishment there of an Islamic court, library, and

research facility similar to the House of Wisdom in Baghdad brought the fruits of medieval Islamic schol- arship to western Europe’s doorstep. However, as Islam found its position on the Iberian peninsula increas- ingly compromised in the twelfth and thirteenth cen- turies, this Islamic learning, as well as some of the ancient Greek scholarship that the medieval Islamic scholars had preserved in Latin translation, began to filter into medieval Europe. In particular, fibonacci [VI.6], son of an influential administrator within the Pisan city state, encountered al-Khw¯arizm¯ı’s text and recognized not only the impact that the Arabic num- ber system detailed there could have on accounting and commerce (Roman numerals and their cumber- some rules for manipulation were still widely in use) but also the importance of al-Khw¯arizm¯ı’s theoretical discussion, with its wedding of geometrical proof and the algorithmic solution of what we can interpret as first- and second-degree equations. In his 1202 book Liber Abbaci, Fibonacci presented al-Khw¯arizm¯ı’s work almost verbatim, and extolled all of these virtues, thus effectively introducing this knowledge and approach into the Latin West.

Fibonacci’s presentation, especially of the practi- cal aspects of al-Khw¯arizm¯ı’s text, soon became well- known in Europe. So-called abacus schools (named after Fibonacci’s text and not after the Chinese calculat- ing instrument) sprang up all over the Italian peninsula, particularly in the fourteenth and fifteenth centuries, for the training of accountants and bookkeepers in an increasingly mercantilistic Western world. The teach- ers in these schools, the “maestri d’abaco,” built on and extended the algorithms they found in Fibonacci’s text. Another tradition, the Cossist tradition—after the German word “Coss” connoting algebra, that is, “Kun- strechnung” or “artful calculation”—developed simul- taneously in the Germanic regions of Europe and aimed to introduce algebra into the mainstream there.

In 1494 the Italian Luca Pacioli published (by now this is the operative word: Pacioli’s text is one of the earliestprinted mathematical texts) a compendium of all known mathematics. By this time, the geometrical justifications that al-Khw¯arizm¯ı and Fibonacci had pre- sented had long since fallen from the mathematical ver- nacular. By reintroducing them in his book, theSumma, Pacioli brought them back to the mathematical fore.

Not knowing of Khayyam’s work, he asserted that solu- tions had been discovered only in the six cases treated by both al-Khw¯arizm¯ıand Fibonacci, even though there had been abortive attempts to solve the cubic and even

though he held out the hope that it could ultimately be solved.

Pacioli’s book had highlighted a key unsolved prob- lem: could algorithmic solutions be determined for the various cases of the cubic? And, if so, could these be justified geometrically with proofs similar in spirit to those found in the texts of al-Khw¯arizm¯ıand Fibonacci?

Among several sixteenth-century Italian mathemati- cians who eventually managed to answer the first ques- tion in the affirmative wascardano[VI.7]. In hisArs Magna, orThe Great Art, of 1545, he presented algo- rithms with geometric justifications for the various cases of the cubic, effectively completing the cube where al-Khw¯arizm¯ı and Fibonacci had completed the square. He also presented algorithms that had been dis- covered by his student Ludovico Ferrari (1522–65) for solving the cases of the quartic. These intrigued him, because, unlike the algorithms for the cubic, they were not justified geometrically. As he put it in his book, “all those matters up to and including the cubic are fully demonstrated, but the others which we will add, either by necessity or out of curiosity, we do not go beyond barely setting out.” An algebra was breaking out of the geometrical shell in which it had been encased.

5 Algebra Is Born

This process was accelerated by the rediscovery and translation into Latin of Diophantus’s Arithmetica in the 1560s, with its abbreviated presentational style and ungeometrical approach. Algebra, as a general problem-solving technique, applicable to questions in geometry, number theory, and other mathematical set- tings, was established in raphael bombelli’s [VI.8]

Algebra of 1572 and, more importantly, in viète’s [VI.9] In Artem Analyticem Isagoge, or Introduction to the Analytic Art, of 1591. The aim of the latter was, in Viète’s words, “to leave no problem unsolved,”

and to this end he developed a true notation—using vowels to denote variables and consonants to denote coefficients—as well as methods for solving equations in one unknown. He called his techniques “specious logistics.”

Dimensionality—in the form of his so-calledlaw of homogeneity—was, however, still an issue for Viète.

As he put it, “[o]nly homogeneous magnitudes are to be compared to one another.” The problem was that he distinguished two types of magnitudes: “lad- der magnitudes”—that is, variables (Aside) (orxin our modern notation), (Asquare) (orx²), (Acube) (orx³),