BRUCE C. BERNDT, BLAIR K. SPEARMAN, KENNETH S. WILLIAMS Commentary on an Unpublished Lecture by G, N, Watson on Solving the Quintic
he following notes are from a lecture on solving quintic equations given by the late Professor George Neville Watson (1886-1965) at Cambridge University in 1948. They were discovered by the first author in 1995 in one of two boxes of papers of Pro- ~ fessor Watson stored in the Rare Book Room of the Library at the University of
Birmingham, England. Some pages that had become sep- 20. In view of the current interest (both theoretical and arated from the notes were found by the third author in computational) in solvable quintic equations [24], [33], one of the boxes during a visit to Birmingham in 1999. [43]-[46], it seemed to the duthors to be of interest to pub- "Solving the quintic" is one of the few topics in mathe- lish Professor Watson's notes on his lecture, with com- matics which has been of enduring and widespread inter- mentary explaining some of the ideas in more current est for centuries. The history of this subject is beautifully mathematical language. For those having a practical need illustrated in the poster produced by MATHEMATICA. for solving quintic equations, Watson's step-by-step pro- Many attempts have been made to solve quintic equations; cedure will be especially valuable. Watson's method ap- see, for example, [6]-[14], [17]-[21], [28]-[32], [34]-[36], plies to any solvable quintic polynomial, that is, any [58]-[60]. Galois was the first mathematician to deter- quintic polynomial whose Galois group is one of Z/5~, D5 mine which quintic polynomials have roots expressible or F2o. in terms of radicals, and in 1991 Dummit [24] gave for- Watson's interest in solving quintics was undoubtedly mulae for the roots of such solvable quintics. A quintic is motivated by his keen interest in verifying Srinivasa solvable by means of radicals ~f and only if its Galois Ramanujan's determinations of class invariants, or group is the cyclic group 77/5W_ of order 5, the dihedral equivalently, singular moduli. Ramanujan computed the group D5 of order 10, or the Frobenius group F2o of order values of over 100 class invariants, which he recorded
The first author thanks Professor Norrie Everitt of the University of Birmingham for an invitation to visit the University of Birmingham in October 1995. The third author thanks Carleton University for a travel grant which enabled him to travel to the University of Birmingham, England in December 1999. The authors thank the staff of the University of Birmingham Library for making the papers of Watson available to them.
9 2002 SPRINGER-VERLAGNEW YORK, VOLUME 24, NUMBER4, 2002 1 5 were verified in two papers by Berndt, Chan, and Zhang [2], [3]; see also Berndt's book [1, Part V, Chapter 34]. Chan [16] has used class field theory to put Watson's determi- nations on a firm foundation, and Zhang [61], [62] has used Kronecker's limit formula to verify Watson's calcu- lations. Professor Watson held the Mason Chair of Mathemat- ics at the University of Birmingham from 1918 to 1951. He was educated at Cambridge University (1904-1908), where he was a student of Edmund Taylor Whittaker (1873-1956). He became a Fellow of Trinity College, Cambridge, in 1910. From 1914 to 1918 he held aca- demic positions at University College, London. Watson devoted a great deal of his research to extending and pro- viding proofs for results contained in Ramanujan's Note- books [39]. He wrote more than thirty papers related to Ramanujan's work, including the aforementioned papers on class invariants or singular moduli. Most mathematicians know Watson as the co-author with E. T. Whittaker of the classic book A Course of Mod- em Analysis, first published in 1915, and author of the monumental treatise Theory of Bessel Functions, first published in 1922. For more details of Watson's life, the reader may wish to consult [22], [41], [48]. We now give the text of Watson 's lecture with our com- mentary in italics. In the course of the text we give the contents of three sheets which presumably Watson handed out to his audience. The first of these gives the basic quan- tities associated with a quintic equation, the second gives twenty-four pentagrams used in showing that permuta- tions of the suffixes of
(XlX 2 + X2X 3 + X3X4 + X4X 5 + X5Xl
-- XlX 3 -- X3X5 -- X5X 2 -- X2X 4 -- X4Xl) 2 yield six distinct expressions, and the third gives Wat- G. N. Watson son's method of solving a solvable quintic equation in radicals. without proofs in his paper [37] and at scattered places throughout his notebooks, especially in his first notebook I am going to begin by frankly admitting that my subject [39]. Although many of Ramanujan's class invariants had this evening is definitely old-fashioned and is rather stodgy; been also calculated by Heinrich Weber [57], most had not you will not fend anything exciting or thrilling about it. been verified. Class invariants are certain algebraic num- When the subject of quintic equations was first seriously bers which are normally very difficult to calculate, and investigated by Lagrange it really was a "live" topic; the ex- their determinations often require solving a polynomial tent of the possibility of solving equations of various de- equation of degree greater than 2; and, in particular, 5. grees by means of radicals was of general interest until it Watson [52] used modular equations in calculating was realized that numbers represented by radicals and some of Ramanujan's class invariants; solving polyno- roots of algebraic equations were about what one nowa- mial equations of degree exceeding 2 was often needed. days calls algebraic numbers. In a series of six further papers [50]-[51], [53]-[56], he de- It is difficult to know quite how much to assume that veloped an empirical process for calculating class in- you already know about solutions of algebraic equations variants, which also depended heavily on solving poly- but I am going to take for granted... nomial equations of high degree. He not only verified Watson's notes do not state the prerequisites for the lecture! several of Ramanujan's class invariants but also found many new ones. For these reasons, Watson proclaimed in I cannot begin without saying how much I value the com- his lecture that he had solved more quintic equations than pliment which you have paid me by inviting me to come any other person. Despite Watson's gargantuan efforts in from a provincial University to lecture to you in Cambridge; calculating Ramanujan's class invariants, eighteen re- and now I am going to claim an 01d man's privilege of in- mained unproven until recent times. The remaining ones dulging in a few reminiscences. In order to make my lec-
16 THE MATHEMATICAL INTELLIGENCER SHEET 1
The denumerate form of the quintic equation is
PoY 5 + PlY 4 § P2Y 3 + P3Y 2 + P4Y + P5 = O.
The standard form of the quintic equation is
ax 5 + 5bx 4 + 10cx 3 + 10dx 2 § 5ex +f= 0. (x -- 10y)
The reduced form of the quintic equation is z 5+ 10Cz 3+ 10Dz 2+5Ez+F=0.(z=ax+b)
The sextic resolvent is a6~b6 - 100Ka4~b 4 + 2000La2r 2 - 800a2~b~/-~ + 40000M = 0,
where
g ~ ae - 4bd + 3c 2, L= -2a2df + 3a2e 2 + 6abcf - 14abde - 2ac2e + 8acd 2 - 4b3f + lOb2ce + 20b2d 2 - 40bc2d + 15c 4, M= a3cf2 - 2a3def + a3e 3 _ a2b2f2 - 4a2bcef + 8a2bd2f - 2a2bde 2 -2a2c2df - lla2c2e 2 + 28a2cd2e - 16a2d 4 + 6ab3ef -12ab2cdf + 35ab2ce 2 - 40ab2d2e + 6abc3f- 70abc2de + 80abcd 3 + 35ac4e - 40ac3d 2 - 25b4e 2 + lOOb3cde -50b2c3e - lOOb2c2d 2 § lOObc4d - 25c 6.
ture effective, I must endeavour to picture to myself what an inspection of the Tripos lists would show that the most is passing in the mind of John Brown who is sitting some- likely person to satisfy the requisite conditions would have where in the middle of this room and who came up to Trin- been the late Lord Rayleigh, who was subsequently Chan- ity last October; and I suppose that, in view of the recent cellor. Probably to you he seems quite prehistoric; to me decision about women's membership of the University, he was an elderly and venerable figure whose acquaintance with the name of John Brown I must couple the name of I made in 1912, and with whom I subsequently had some his cousin Mary Smith who came up to Newnham at the correspondence about electric waves. You cannot help re- same time. To try to read their thoughts I must cast my garding me as equally elderly, but I hope that, for a num- mind back 43 years to the Lent Term of 1905 which was in ber of reasons you do not consider me equally venerable, my first year. If I had then attended a lecture by a mathe- and that you will believe me when I say that I still have a matician 43 years my senior who was visiting Cambridge, good deal of the mentality of the undergraduate about me. However, so far as I know, Lord Rayleigh did not visit Cam- bridge in the Lent term of 1905, and so my attempt at an anal- ogy rather breaks down. On the other hand a visit was paid to Cambridge at the end of that term by a much more emi- nent personage, namely the Sultan of Zanzibar. For the ben- efit of those of you who have not heard that story, I mention briefly that on the last day of term the Mayor of Cambridge received a telegram to the effect that the Sultan and his suite would be arriving by the mid-day train from King's Cross and would be glad if the Mayor would give them lunch and arrange for them to be shown over Cambridge in the course of the afternoon. The program was duly carried out, and during the next few weeks it gradually emerged that the so-called Sul- tan was W. H. de Vere Cole, a third-year Trinity undergradu- ate. It was the most successful practical joke of an age in Watson Building, University of Birmingham. (Photo from 1995.) which practical joking was more popular than it is to-day.
VOLUME 24, NUMBER 4, 2002 1 7 If you could be transported back to the Cambridge of insularity, if I may so describe it, of my education. If that 1905, you would find that it was not so very different from hypothetical lecture by Lord Rayleigh had taken place, he the Cambridge of 1948. One of the differences which would could have given a more striking illustration of insularity strike you most would probably be the fact that there were which you will probably hardly credit. In his time, each Col- very few University lectures (and those mostly professor- lege tutor was responsible for the teaching of his own ial lectures which were not much attended by undergrad- pupils and of nobody else; he was aided by one or two as- uates); other lectures were College lectures, open only to sistant tutors, but the pupils, no matter what subject they members of the College in which they were given, or, in were reading, received no official instruction except from the case of some of the smaller Colleges, they were open their own tutor and his assistants. to members of two or three colleges which had associated After spending something like ten minutes on these ir- themselves for that purpose. Thus most of the teaching relevancies, it is time that I started getting to business. which I received was from the four members of the Trin- There is one assumption which I am going to make ity mathematical staff; the senior of them was Herman, who throughout, namely that the extent of your knowledge died prematurely twenty years ago; in addition to teaching about the elements of the theory of equations is roughly me solid geometry, rigid dynamics and hydrodynamics, he the same as might have been expected of a similar audi- infected me with a quality of perseverance and tenacity of ence in 1905. For instance, I am going to take for granted purpose which I think was less uncommon in the nine- that you know about symmetric functions of roots in terms teenth century than it is to-day when mathematics is tend- of coefficients and that you are at any rate vaguely famil- ing to be less concrete and more abstract. Whitehead was iar with methods of obtaining algebraic solutions of qua- still alive when I started collecting material for this lecture. dratic, cubic and quartic equations, and that you have heard Whittaker, who lectured on Electricity and Geometric Op- of the theorem due to Abel that there is no such solution tics, whose name is sometimes associated with mine, is liv- of the general quintic equation, i.e., a solution expressible ing in retirement in Edinburgh; and Barnes is Bishop of by a number of root extractions. Birmingham. Outside the College I attended lectures by Baker on Theory of Functions, Berry of King's who taught In modern language, if f(x) E Q[x] is irreducible and of de- me nearly all of what I know of elliptic functions, and Hob- gree 5, then the quintic equation f(x) = 0 is solvable by rad- son on Spherical Harmonics and Integral Equations; also icals if and only if the Galois group G off(x) is solvable. two of the Professors of that time that were Trinity men, The Galois group G is solvable if and only if it is a sub- Forsyth and Sir George Darwin, whom I remember lectur- group of the Frobenius group F20 of order 20, that is, it is ing on curvature of surfaces and the problem of three bod- F20, D5 (the dihedral group of order 10), or E/5E (the cyclic ies, respectively. Two things you may have noticed, the group of order 5); see for example [24, Theorem 2, p. 397], large proportion of my teachers who are still alive, and the [25, Theorem 39, p. 609]. Thus a quintic equation f(x) = 0 cannot have its roots expressed by a finite number of root extractions if the Galois group G off is non-solvable, that is, if it is $5 (the symmetric group of order 120), or A5 (the alternating group of order 60). "Almost all" quintics have $5 as their Galois group, so the "general" quintic is not solv- able by radicals. It is easy to give examples of quintics which are not solvable by radicals; see for example [46].
You may or may not have encountered the theorem that any irreducible quintic which has got an algebraic solution has its roots expressible in the form
U 0 4- tOrUl 4- o)2ru 2 4- oj3ru3 4- w4ru4,
where ~o denotes exp(2~v//5), r assumes the values 0, 1, 2, 3, 4, and u 5, u 5, u 5, u 5 are the roots of a quartic equation whose coefficients are rational functions of the coefficients of the original quintic. If you are not familiar with such re- sults, you will find proofs of them in the treatise by Burn- side and Panton.
One can find this in Section 5 of Chapter XX of Burnside and Panton's book [5, Vol. 2]. A modern reference for this result is [24, Theorem 2, p. 397].
When I was an undergraduate, all other knowledge about Bruce Berndt with Ramanujan's Slate. quintic equations was hidden behind what modern politi-
18 THE MATHEMATICAL/NTELLIGENCER cians would describe as an iron curtain, and it is conve- with the property that their cubes have, not six different nient for me to assume that this state of affairs still per- values, but only two, namely sists, for otherwise it would be a work of a supererogation (~ +/3e + 3'e2)~, (~ + 13~2 + 3'~)3, for me to deliver this lecture. I might mention at this point that equations of the fifth and these expressions are the roots of a quadratic equation or a higher degree which possess algebraic solutions (such whose coefficients are rational functions of the coefficients equations are usually described as Abelian) are of some im- of the cubic. When the cubic equation is portance in the theory of elliptic functions, apart from their ax 3 + 3bx 2 + 3cx + d = O, intrinsic interest. the quadratic equation is Today such equations are called solvable. a6X 2 + 27a3(a2d - 3abc + 2b3)X + 729(b 2 - ac) 3 = O,
and there is no difficulty in completing the solution of the There is, for instance, a theorem, also due to Abel, that the cubic. equations satisfied by the so-called singular moduli of el- liptic functions are all Abelian equations. It is easily checked using MAPLE that this quadratic is correct. Singular moduli are discussed in Cox's book [23, Chapter 3] as well as in Berndt's book [1, Part V, Chapter 34]. For the quartic equation, with roots a,/3, T, 8, such ex- pressions as It was these singular moduli which aroused my interest some fifteen years ago in the solutions of Abelian equa- (Ol-F /3-- 3'-- 8) 2, (O~A-~-- t~-- /3)2, (OlA- 8/" /3-- 3') 2 tions, not only of the fifth degree, but also of the sixth, sev- have only three distinct values; similar but slightly simpler enth and other degrees higher still. It consequently became expressions are necessary for me to co-ordinate the work of previous writ- ers in such a way as to have handy a systematic procedure a/3 + 3'8 - a3' - a8 -/33, -/38, etc., for solving Abelian quintic equations as rapidly as possible, or simpler still, and this is what I am going to describe tonight. ~/3 + 3'8, aT +/38, ~8 +/33'. Methods for solving a general solvable quintic equation When the quartic equation is taken to be in radicals have been given in the 1990s by Dummit [24] ax 4 + 4bx 3 + 6cx 2 + 4dx + e = 0, and Kobayashi and Nakagawa [33]; see also [47]. the cubic equation satisfied by the last three expressions is To illustrate the nature of the problem to be solved, I aX 8 - 6a2cX 2 + (16bd - 4ae)aX am now going to use equations of degrees lower than the - (16b2e + 16ad 2 - 24ace) = 0, fifth as illustrations. A reason why such equations pos- sess algebraic solutions (and it proves to be the reason) and, by the substitution is that certain non-symmetric functions of the roots ex- aX- 2c = -40, ist such that the values which certain powers of them can assume are fewer in number than the degree of the equa- this becomes tion. Thus, in the case of the quadratic equation with 403 - O(ae - 4bd + 3c 2) roots c~ and/3, there are two values for the difference of - (ace + 2bcd - ad 2 - b2e - c 3) = 0, the roots, namely which is the standard reducing cubic o~-/3,/3 - ,:~. 403 - IO- J= O. However the squares of both of these differences have one This is discussed in [26, pp. 191-197; see problem 15, p. value only, namely 197], where the values of I and J are given by (a +/3) 2 - 4aft, ab! and this is expressible rationally in terms of the coeffi- I=ae-4bd+3c 2, J= b c cients. Hence the values of the differences of the roots are c d obtainable by the extraction of a square root, and, since the I have discussed the problem of solving the quartie equa- sum of the roots is known, the roots themselves are im- tion at some length in order to be able to point out to you mediately obtainable. the existence of a special type of quartic equation which The cubic equation, with roots a,/3, y, can be treated sim- rarely receives the attention that it merits. In general the re- ilarly. Let e3 = 1 (e r 1). Then we can form six expressions ducing cubic of a quartic equation has no root which is ra- a+/3E+ye 2, ~+yE+~2, ~/+ae+/3e z, tional in the field of its coefficients, and any expression for a+/3e2+Te, /3+Te2+~, T+ae2+fle, the roots of the quartic involves cube roots; on the other
VOLUME 24, NUMBER 4, 2002 19 SHEET 1A
The discriminant A of the quintic equation in its standard form is equal to the product of the squared differences of the roots multiplied by a8/3125. The value of the discriminant A in terms of the coefficients is
a4f4 - 20a3bcf3 - 120a3cdf3 + 160a3ce2f 2 + 360a3d2ef 2 -640a3de3f + 256a3e 5 + 160a2b2df 3 - lOa2b2e2f2 + 360a2bc2f3 - 1640a2bcdef2 + 320a2bce3f - 1440a2bd3f2 +4080a2bd2c2f- 1920a2bde 4 - 1440a2c3ef2 + 2640a2c2d2f 2 + 4480a2c2de2f - 2560a2c2e 4 - l O080a2cdUef + 5760a2cd2e 3 + 3456a2d5f - 2160a2d4e 2 - 640ab3cf3 + 320ab3def 2 -180ab3c3f + 4080ab2c2ef 2 + 4480ab2cd2f 2 - 14920ab2cde2f + 7200ab2ce 4 + 960ab2d3ef- 600ab2d2e 3 - lO080abcadf2 +960abcae2f+ 28480abc2d2ef - 16000abc2de 3 - 11520abcd4f + 7200abcdUe 2 + 3456ac5f 2 - 11520ac4def + 6400ac4e 3 +5120ac3d3f - 3200ac3d2e 2 + 256b5f 3 - 1920b4cef2 -2560b4d2f 2 + 7200b4de2f - 3375b4c 4 + 5760b3c2df 2 -600bac2e2f- 16000b3cd2ef + 9000b3cde 3 + 6400b3d4f -4000bSd3e 2 - 2160b2c4f2 + 7200b2cadef - 4000b2c3e3 - 3200b2c2daf + 2000b2c2d2e 2.
hand, there is no difficulty in constructing quartic equations I shall show when I say anything derived from the theory of whose reducing cubics possess at least one rational root; the groups. On the other hand, while to the best of my knowl- roots of such quartics are obtainable in forms which involve edge nobody else has solved more than about twenty Abelian the extraction of square roots only. Such quartics are anal- quintics (you will be hearing later about these solvers, and I ogous to Abelian equations of higher degrees, and it might have no certain knowledge that anybody else has ever solved be worth while to describe them either as "Abelian quartic any), my own score is something between 100 and 120; and equations" or as "biquadratic equations," the latter being an I must admit that I feel a certain amount of pride at having alternative to the present usage of employing the terms quar- so far outdistanced my nearest rival. tic and biquadratic indifferently. (I once discussed this ques- tion with my friend Professor Berwick, who in his lifetime Young solved several quintic equations in [58] and [59]. was the leading authority in this country on algebraic equa- tions, and we both rather reluctantly came to the conclusion The notation which I use is given at the top of the first that the existing terminology was fixed sufficiently firmly to of the sheets which have been distributed. The first equa- make any alteration in it practically impossible.) tion, namely
PoY 5 + PlY a + P2Y 3 + P3Y 2 + P4Y + P5 = O, If f(x) E Q[x] is an irreducible quartic polynomial, its cubic resolvent has at least one rational root if and only is what Cayley calls the denumerate form, while if the Galois group off(x) is the Klein 4-group V4 of or- ax 5 + 5bx 4 + 10cx3 + 10dx 2 + 5ex + f = O, der 4, the cyclic group ~_/4~_ of order 4, or the dihedral is the standard form. The second is derived from the first group D4 of order 8. Since D4 is not abelian, it is not ap- by the substitution 10y = x, with the relations propriate to call such quartics "abelian. "For the solution of the quartic by radicals, see for example [25, p. 548]. a=po, b=2pl, c=10p2, d=100P3, e = 2000p4, f = 105p5. After this very lengthy preamble, I now reach the main Next we carry out the process usually described as "re- topic of my discourse, namely quintic equations. Some of you moving the second term" by the substitution ax + b = z, may be familiar with the name of William Hepworth Thomp- which yields the reduced form son, who was Regius Professor of Greek from 1853 to 1866, and subsequently Master of Trinity until 1886. A question was z 5 + 10Cz 3 + 10Dz 2 + 5Ez + F = O, once put to him about Greek mathematics, and his reply was, in which "I know nothing about the subject. I have never even lectured C = ac - b 2, D = a2d - 3abc + 2b 3, upon it." Although there are large tracts of knowledge about E = aae - 4a2bd + 6ab2c - 3b 4, quintic equations about which I am in complete ignorance, I F = a4f- 5aabe + lOa2b2d - lOabac + 4b 5. have a fair amount of practical experience of them. For in- stance, if my friend Mr. P. Hall of King's College is here this The roots of the last two quintics will be denoted by Xr evening, he will probably be horrified at the ignorance which and Zr respectively with r = 1, 2, 3, 4, 5.
20 THE MATHEMATICALINTELUGENCER ,. / ~ ~ /" ~ A ~ A ~ ~ A ~ ~ A =
-
E SHEET 3
The roots of the quintic in its reduced form are
Zr+ 1 = o)rUl 4- o)2ru 2 4- e)3ru3 "4- o)4ru 4
with w = exp(2~r//5), r = 1, 2, 3, 4, 0.
(1) UlU4 + u2u3 = -2C. (2) + U Ul + u u4 + = -2D. (3) 2 2 u 2 2 u3u2 3 u3ul-u~u3=E. UlU 4 4- 2U3 -- UlU2U3U 4 -- __ U2U4 -- (4) u5[+ u 5 + u 5 + u 5 - 5(ulu4 - u2us)(u21u3 - u2ul - u2u4 + u2u2) = -F.
New unknowns, 0 and T, defined by
(5) UlU4 - u2u3 = 20, (6) u u3 + - - u u4 = 2T. UlU4 = -C 4- 0, u2u 3 = -C - O. u2u3 + u2u2 = -D + T, U2Ul 4- u2u4 = -D - r. u2u3 - u2u2 = -+ ~/(D - T) 2 + 4(C - 0)2(C + 0) = RI, u~2ul - u2u4 -- -+ ~/(D + T) 2 + 4(C + 0)2(C - 0) = R2. u~u2 = (u2u3)(u2ut)/(u2u3), etc., u 5 = (u2u3)2(u2ul)/(u2u3) 2, etc. (7) C(D 2 - T 2) + (C 2 - 02)(C 2 + 302 - E) = R1R20. (8) (D 2 - T2) 2 + 2C(D 2 - T2)(C 2 + 302) - 8C02(D 2 + T 2) +(C 2 - 02)2(C2 - 502)2 + 16DO3T + E2(C 2 - 02) - 2CE(D 2 - T 2) - 2E(C 2 - 02)(C 2 + 302) = 0. (9) (DO + CT)(D 2 - T 2) + T(C 2 - 502) 2 - 2CDEO -ET(C 2 + 02) + Fe(C 2 - 02) = 0.
Young's substitutions are
T = Ot, 02 = ~.
The connexion between the 0 above and the r of Cayley's sextic resolvent is
100~/-5 = a2r
The denumerate quintic of Ramanujan's problem is
y5 _ y4 + y3 _ 2y2 + 3y - 1 = 0.
For this quintic, C = 6, D = -156, E = 4592, F = -47328.
z = 10y - 2, 0 = -10~v~, t = -10, T = 100~-5.
u 5, u 5 = -13168 - 6400~x/5 -+ (2160 + 960~/-5)X/79(5 - 2~/5),
u 5, u 5 = -13168 + 6400X/-5 -+ (2160 - 960%/5)N/79(5 + 2~/5).
R1,2 R22_ 79(800-+ 160V5), RIR2 = -320 x 79~.
We remark that Young's equations for t and ~ in this and example are: (-156 + 6t)(24336 - Ot 2) + t(36 - 5~) 2 (24336 - Ot 2) + 12(24336 - 0t2)(36 + 30) + 6892416 - 4592t(36 + 0) + 473280 = 0,
- 48~(24336 + 0t 2) + (36 - 0)(36 - 50) 2 so that t =-10 and 0= 500 in agreement with
- 2496~b2t - 581898240 - 21086464~ Watson. + 551040t 2 - 9184(36 - 0)(36 + 3~) = 0
22 THE MATHEMATICAL INTELLIGENCER Our next object is the determination of non-symmetric all possible orders (there is no loss of generality in taking functions of the roots which can be regarded as roots of a the number 1 in a special place) and the rule for determi- resolvent equation. An expression which suggests itself is nation of signs is that terms associated with adjacent ver- tices are assigned + signs, while those associated with op- (Xl § COX2 § tf2X3 § tO3X4 § tO4X5)5. posite vertices are assigned - signs. The result of permuting the roots is to yield 24 values for Now the pentagrams in the third and fourth columns are the expression. the optical images in a vertical line of the corresponding pentagrams in the first and second columns, and since A permutation (r ~ $5 acts on this element by proximity and oppositeness are invariant for the operation of taking an optical image, the number of distinct values of of(x, + ~x2 + ~o2x3 + ~03Xn + ~04x5)5) & is reduced from 24 to 12. : (Xr § tOX(r(2) § tO2X~(3) § tO3X~(4) § tO4X(r(5)) 5" Further, the pentagrams in the second column are de- An easy calculation shows that (r preserves rived from those in the first column by changing adjacent vertices into opposite vertices, and vice versa, so that the OL : (X 1 § t0X2 § tO2X3 § t03X4 § 0)4X5) 5 values of ~b arising from pentagrams in the second column if and only if (r = (1 2 3 4 5)k for some k E {0, 1, 2, 3, 4}. are minus the values of ~b arising from the corresponding Hence pentagrams in the first column. It follows that the number of distinct values of (b2 is not 12 but 6, and so our resolvent stabs5(a) = ((1 2 3 4 5)>, has now been reduced to a sextic equation in r with co- so that efficients which are rational functions of the coefficients of the quintic, and a sextic equati.on is a decided improvement Istabss(a)l = 5. on an equation of degree/20, or even on one of degree 24. Thus, by the orbit-stabilizer theorem [27, p. 139], we obtain Let a = (12345) E $5 and b -- (25)(34) ~ $5, so that a 5 = iorbs5(a)l = ~ _ 120 _ 24, b 2 = e and bab = a 4. As acbl = c~1 and bob1 = ~bl, we have 5 5 stabss(r ) >- (a, bla 5 = b 2 = e, bab = a 4> = Ds, so that permuting the roots yields 24 different expressions. so that The disadvantage of the corresponding resolvent equation Istabs5(~b,)l-> 10. is the magnitude of the degree of its coefficients when ex- pressed as functions of the coefficients of the quintic; more- On the other hand, the first two columns of Watson's pen- over it is difficult to be greatly attracted by an equation tagram table show that whose degree is as high as 24 when our aim is the solution ]orbs5(~bl)l-> 12. of an equation of degree as low as 5. An expression which is more amenable than the ex- Hence, by the orbit-stabilizer theorem, we see that pression just considered was discovered just 90 years ago Istabs5(r = 10, Iorbs5(~bl)l = 12 by two mathematicians of some eminence in their day, namely Cockle and Harley, and it was published in the and thus Memoirs of the Manchester Literary and Philosophical So- stabs5(~bl) = (a, bla 5 = b 2 = e, bab = a 4> = D5. ciety. This expression is Now let c = (2 3 4 5), so that c 2 = b. As a4~2 = r and cr ~1 = XlX2 § X2X3 § X3X4 § X4X5 § X5Xl -- XlX3 = cb2, we have -- X3X5 -- X5X2 -- X2X4 -- X4Xl. stabs5(~b 2) D_ (a, cla 5 = c 2 = e, c lac = a 3> = F20, The quantities xlx2 + x2x3 + x3x4 § x4x5 § X5Xl and XlX3 + X3X5 + X5X2 § X2X 4 § X4X 1 appear in the work of Harley so that Istabs5(~b2)l --- 20. From the first column of the pen- [29] and their difference is considered by Cayley [6]. We tagram table, we have have not located a joint paper of Cockle and Harley. When Iorbsa(r 6. he was writing these notes, we believe Watson was read- ing from Cayley [6] where the names of Cockle and Harley Hence, by the orbit-stabilizer theorem, we deduce that are linked [6, p. 311]. Istabs5(r = 20, Iorbs5((,b2)l = 6,
Permutations of the suffixes give rise to 24 expressions, and thus which may be denoted by ~ _+ X~s, where r and s run stabs5(~b 2) = (a, c]a 5 = c a = e, c lac = a n} = F20. through the values 1, 2, 3, 4, 5 with r r s. The choice of the signs is most simply exhibited diagrammatically, with each It is, however, possible to effect a further simplification; of the 24 expressions represented by a separate diagram. it is not, in general, possible to construct a resolvent equa- If you turn to the second page of your sheets, you will see tion of degree less than 6, but it is possible to construct a the 24 pentagrams with vertices numbered 1, 2, 3, 4, 5 in sextic resolvent equation in which two of the coefficients
VOLUME 24, NUMBER 4, 2002 23 are zero. We succeeded in constructing a sextic in 4)2 be- If r is odd, the transposition (12) changes the above equa- cause the 12 values of r could be grouped in pairs with the tion to members of each pair numerically equal but opposite in -Er = (X2 -- xl)q(xl, x3, x4, x5) + r(xl, x3, x4, x5). sign; but a different grouping is also possible, namely a se- lection of one member from each of the six pairs so as to Adding these two equations, we obtain form a sestet in which the sum of the members is zero, and 0 = (xl - x2)(q(x2, x3, x4, x5) - q(xl, x3, x4, x5)) it is evident that those members which have not been se- + r(x2, x3, x4, x5) + r(xl, x3, x4, x5). lected also form a sestet in which the sum of the members is zero; one of these sestets is represented by the penta- Taking Xl = x2 we deduce that grams in the first column, the other by the pentagrams in r(x2, x3, x4, x5) = 0. the second column. Hence A sestet is a set of six objects. Er ---- (Xl -- x2)q(x2, 9 9 9 , x5).
Denote the values of (5 represented by the pentagrams Thus Xl - x2 divides Er. Similarly Xm - xn divides Er for in the first column by (51, ~, 9 9 9 (56, and let m, n = 1, 2, 3, 4, 5, m < n. Hence
(5~ + (5~ + " " " + (5~= Er.
[I (xm - Xn) It is then not difficult to verify that an interchange of any l~
po'O E stabs5(51 = D5 C A5. These properties ensure that the polynomial Hence o-E A5, which is a contradiction. Now ((5 - (51)((5 - ~)"" ((5 - (56)
{0((5t),..., 0((56)} C orbs5(51, I0rb35(511 = 12, has coefficients in Q or Q(V~), where D is the discrim- inant of the quintic, with the coefficients of (55 and (53 and equal to zero.
{0((51), 9 9 9 0((5~)} n {(51, 9 9 9 , (56} = O, Apart from the graphical representation by pentagrams so that (which, as the White Knight would say, is my own inven-
{0~(51),'" ", 0~(56)} = {--(51,''', --(56}" tion), all of the analysis which I have just been describing was familiar to Cayley in 1861; and he thereupon set about Thus if "r E $5 is a transposition, the construction of the sextic resolvent whose roots are r(Er) = r((5[ + ... + (5~) (51, (52, 9 9 9 (56. The result which he obtained was the equa- = (-- (51) r 4- " " " + (-- (56) r ---- (-- 1) r Er. tion
It is now evident that each of the 10 expressions Xm - Xn (0) a6(56 - 100Ka4r 4 + 2000La2r 2 (m, n = 1, 2, 3, 4, 5; m < n) is a factor of E~ whenever r is - 800a2(5~5A + 40000M = 0 an odd integer. in which the values of K, L, M in terms of the coefficients of the quintic are those given on the first sheet, while A is Clearly Er E 7/[Xl, . . . , x5] and so can be regarded as a the discriminant of the quintic in its standard form, that is polynomial in xt with coefficients in 7/[x2, . . ., x5]. Di- to say, it is the product of the squared differences of the viding Er by xt - x2, we obtain roots of the quintic multiplied by aS/3125. Its value, in terms
Er = (Xl -- x2)q(x2, 9 9 9 , x5) + r(x2, . . . , x5), of the coefficients occupies the lower half of the first sheet. where The work of Cayley to which Watson refers is contained q(x2 .... , x5), r(x2 .... , x5) E 7/[x2,... , x5]. in [6], where on pages 313 and 314 Cayley introduces the
24 THE MATHEMATICAL INTELLIGENCER pentagrams described by Watson. Note that the usual dis- by criminant D of the quintic is [26, p. 205] aS(x1 - x2)2(x1 - x3) 2" 9 9 (x4 - x5) 2 = 3125A = 55A. f(z) = e-~/24~7(~ -~)
There is no obvious way of constructing any simpler re- V(z) solvent and so it is only natural to ask "Where do we go A result equivalent to Ramanujan's assertion was first from here?" It seems fruitless to attempt to obtain an al- proved by Russell [42] and later by Watson [53]; see also gebraic solution of the general sextic equation; for, if we [54]. The solution of this quintic in radicals is given in could solve the general sextic equation algebraically, we [49]. In [38], Ramanujan also posed the problem of find- could solve the general quintic equation by the insertion of ing the roots of another sextic polynomial which factors a factor of the first degree, so as to convert it into a sextic into x - 1 and a quintic satisfied by G47. For additional equation. In this connection I may mention rather a feeble comments and references about this problem, see [4] and joke which was once perpetrated by Ramanujan. He sent [40, pp. 400-401]. Both Weber and Ramanujan calculated to the Journal of the Indian Mathematical Society as a over 100 class invariants, but for different reasons. Class problem for solution: invariants generate Hilbert class fields, one of Weber's Prove that the roots of the equation primary interests. Ramanujan used class invariants to x6 - x3 4- x2 4- 2x- 1 = 0 calculate explicitly certain continued fractions and prod- ucts of theta functions. can be expressed in terms of radicals. After this digression, let us return to the sextic resol- This problem is the first part of Question 699 in [38]. It vent; it is the key to the solution of the quintic in terms of can be found in [40, p. 331]. A solution was given by Wat- radicals, provided that such a key exists. It is l~ossible, by son in [49]. It seems inappropriate to refer to this prob- accident as it were, for the sextic resolvent to have a so- lem as a "feeble joke." lution for which 4)2 is rational, and the corresponding value of 4) is of the form p~f5h, where p is rational. A knowledge Some years later I received rather a pathetic letter from of such a value of 4) proves to be sufficient to enable us to a mathematician, who was anxious to produce something express all the roots of the quintic in terms of radicals. In worth publication, saying that he had noticed that x 4- 1 fact, when this happy accident occurs, the quintic is was a factor of the expression on the left, and that he Abelian, and when it does not occur, the quintic is not wanted to reduce the equation still further, but did not see Abelian. how to do so. My reply to his letter was that the quintic e~luation If 4)2 E Q it is clear from the resolvent sextic that 4) = x 5-x 44-x 3-2x 24-3x- 1 =0 pV'5-h for some p E (~. We are not aware of any rigorous direct proof in the classical literature of the equivalence was satisfied by the standard singular modulus associated of 4)2 E Q to the original quintic being solvable. with the elliptic functions for which the period iK'/K was equal to ~, and consequently it was an Abelian quin- This is as far as Cayley went; he was presumably not in- tic, and therefore it could be solved by radicals; and I told terested in the somewhat laborious task of completing the him where he would find the solution in print. I do not details of the solution of the quintic after the determina- know why Ramanujan inserted the factor x 4- 1; it may tion of a root of his sextic resolvent. have been an attempt at frivolity, or it may have been a The details of the solution of an Abelian quintic were desire to propose an equation in which the coefficients worked out nearly a quarter of a century later by a con- were as small as possible, or it may have been a combi- temporary of Cayley, namely George Paxton Young. I shall nation of the two. not describe Young as a mathematician whose name has been almost forgotten, because he was not in fact a pro- On pages 263 and 300 in his second Notebook [39], Ra- fessional mathematician at all. The few details of his ca- manujan indicates that 21/4G79 is a root of the quintic reer that are known to me are to be found in Poggendorfs equation x 5 - x 4 + x 3 - 2x 2 4- 3x - 1 = 0; see [1, Part V, biographies of authors of scientific papers. He was born in p. 193]. For a positive integer n, Ramanujan defined Gn 1819, graduated M.A. at Edinburgh, and was subsequently by Professor of Logic and Metaphysics at Knox College, Toronto; he was also an Inspector of Schools, and subse- Vn = 2- t/4 f(~/E-n-n), quently Professor of Logic, Metaphysics and Ethics in the where, for any z = x 4- iy E C with y > O, Weber's class University of Toronto. He died at Toronto on February 26, invariant f(z) [57, Vol. 3, p. 114] is defined in terms of the 1889. His life was thus almost coextensive with Cayley's Dedekind eta function (born August 16, 1821, died January 26, 1895). Young in the last decade of his life (and not until then) published a num- ~7(z) = e #iz/12 ~ (1 - e 2~mz) ber of papers on the algebraic solution of equations, in- cluding three in the American Journal of Mathematics
VOLUME 24, NUMBER 4, 2002 25 which among them contain his method of solving Abelian (2) u21u3 ~- U2Ul + u2u4 ~- u2u2 = -2D, quintics. (3) UlU422 ~- u2u2 23 -- UlU2U3U4 -- u iu2 -- u u4 -- u u,
- u3u3 = E,
These are papers [58], [59] and [60]. (4) U 5 -{- U 5 -I- U 5 ~- U 5 -- 5(UlU 4 -- U2U3)(U21U3 -- U2Ul - u2u4 + u2u2) = -F. In style, his papers are the very antithesis of Cayley's. While Cayley could not (or at any rate frequently did not) write These coefficients were essentially given by Ramanu- grammatical English, he always wrote with extreme clar- jan in his first Notebook [39]; see Berndt [1, Part IV, p. 38]. ity, and, when one reads his papers, one cannot fail to be They also occur in [43]. impressed by the terseness and lucidity of his style, by the We next introduce two additional unknowns, 0 and T, mastery which he exercises over his symbols, and by the defined by the equations feeling which he succeeds in conveying that, although he (5) ulua - u2u3 = 28, may have frequently suppressed details of calculation, the (6) u2u3 + u24u2 - u2u, - u2u4 = 2T, reader would experience no real difficulty in filling in the lacunae, even though such a task might require a good deal in which a kind of skew symmetry will be noticed. The nat- of labour. ural procedure is now to determine ul, u2, u3, u4 in terms of On the other hand, when one is reading Young's work, 8, Tand the coefficients of the reduced quintic by using equa- it is difficult to decide what his aims are until one has tions (1), (2), (5) and (6) only. When this has been done, sub- reached the end of his work, and then one has to return to stitute the results in (3) and (4), and we have reached the the beginning and read it again in the light of what one has penultimate stage of our journey by being confronted with discovered; his choice of symbols is often unfortunate; in two simultaneous equations in the unknowns 0 and T. fact when I am reading his papers, I find it necessary to From (1) and (5) we have make out two lists of the symbols that he is using, one list UlU 4 = -C-~ 8, U2U3 = -C- 8, of knowns and the other of unknowns; finally, his results seemed to be obtained by a sheer piece of good fortune, while from (2) and (6) we have and not as a consequence of deliberate and systematic u2u3+u2u2=-D+T, u~ul+u2u4=-D-T; strategy. A comparison of the writings of Cayley and Young shows a striking contrast between the competent draughts- and hence it follows that manship of the lawyer and pure mathematician on the one hand and the obscurity of the philosopher on the other. u2u3 - u2u2 = • ~/(D - T) 2 + 4(C - 0)2(C + 8) =: R1, say; The rest of my lecture I propose to devote to an account u2ul - u2u4 = • ~/(D + T) 2 + 4(C + 0)2(C - 8) =: R2, say. of a practical method of solving Abelian quintic equations. The method is in substance the method given by Young, Watson makes use of the identities but I hope that I have succeeded in setting it out in a more intelligible, systematic and symmetrical manner. (U2U3 -- U2U2)2 = (U2U3 + U2U2) 2 -- 4(UlU4)2(U2U3),
Take the reduced form of the quintic equation (U2Ul -- U2U4) 2 ---- (U2UI -~ U2U4) 2 -- 4(u2u3)2(UlU4).
z 5 + 10Cz 3 + 10Dz 2 + 5Ez + F = O, These last equations enable us to obtain simple expres- sions for the various combinations of the u's which occur and suppose that its roots are in (3) and (4). Thus, in respect of (3), we have Zr+ l z o)rul ~- (.02ru2 -{- o)3ru3 Jr to4ru4 '
U lU2 _ ulu , where U2U3 3 3 w = exp(2qr//5), r = 1, 2, 3, 4, 0. with similar expressions for u2u4, u3ul, u3u3. When we sub- stitute these values in (3) and perform some quite straight- Straightforward but somewhat tedious multiplication forward reductions, we obtain the equation shows that the quintic equation with these roots is (7) C(D 2 - T 2) + (C 2 - 02)(C 2 + 302 - E) = RIR20. Z 5 -- 5Z3(UlU4 -{- U2U3) -- 5Z2(U2U 3 -+- U2Ul "1- U2U4 -}- U2U2) +5z(u~u~ + U2U2 2-3 ulu2u3u4 - U31U2- U32U4- U3Ul This shows incidentally that, when 0 and T have been de- - u u3) termined, the signs of R1 and R2 cannot be assumed arbi- trarily but have to be selected so that R1R2 has a uniquely de- --(U 5 "1- U 5 -+" U 5 -t- U 5 -- 5U3U3U4 -- 5U3UlU3 -- 5U3U4U2 -- 5U3U2U1 terminate value. The effect of changing the signs of both R1 =1=5UlU3U 22 4 -~- 5U2U2U2+ 5U3U2U2 -~ 5U4U2U 2) = 0; and R2 is merely to interchange ul with u4 and u2 with u3. The result of rationalising (7) by squaring is the more and a comparison of these two forms of the quintic yields formidable equation four equations from which ul, u2, Us, u4 are to be deter- mined, namely (D 2 - T2)2 + 2C(D 2 - T2)(C 2 + 302) - 8C02(D2 + T 2) (8) +(C 2 - 02)(C 2 - 502) 2 + 16DO3T + E2(C 2 - 82) (1) ulu4 + u2u3 = -2C, -2CE(D 2 - T 2) - 2E(C 2 - 02)(C 2 + 382) = 0.
2{} THE MATHEMATICAL INTELLIGENCER This disposes of (3) for the time being, and we turn to Young goes on to suggest that, in numerical examples, (4). The formulae which now serve our purpose are his pair of simultaneous equations should be solved by in- spection. He does, in fact, solve the equations by inspec- u 5 = (u2u3)2(u2ul) etc., (u2u3)2 ' tion in each of the numerical examples that he considers, and, although he says it is possible to eliminate either of with three similar formulae. When these results are inserted the unknowns in order to obtain a single equation in the in (4) and the equation so obtained is simplified as much other unknown, he does not work out the eliminant. You as possible, we have an equation which I do not propose will probably realize that the solution by inspection of a to write down, because it would be a little tedious; it has pair of simultaneous equations of so high a degree is likely a sort of family resemblance to (7) in that it is of about the to be an extremely tedious task, and you will not be mis- same degree of complexity and it involves the unknowns 0 taken in your assumption. Consequently Young's investi- and T and the product RIR2 rationally. gations have not got the air of finality about them which could have been desired. MAPLE gives the equation as Fortunately, however, the end of the story is implicitly told in the paper by Cayley on the sextic resolvent which (1) 2 - T2)(DO 2 + 2CT0 + C2D) + 2(C 2 - 02)(3CD02 - TO3) I have already described to you and which had been pub- -RIR2(TO 2 + 2CD0 + C2T) + (C 2 - 02)2(20T0 - F) = O. lished over a quarter of a century earlier. It is, in fact, easy When we substitute for this product R1R2 the value which to establish the relations is supplied by (7), we obtain an equation which is worth z~z2 + .... ZlZ3 .... writing out in full, namely = a2(xlx2 + .... xlx3 .... ) = a2r
(DO + CT)(D 2 - T 2) + T(C 2 - 502) 2 - 2CDEO and also to prove that the~xpression on the left is equal to (9) -ET(C 2 + 02) + FO(C 2 - 02) = O. 5(ulu4 - u2u~)X/5 We now have two simultaneous equations, (8) and (9), in which the only unknowns are 0 and T. When these equations so that have been solved, the values of Ul, u2, u3, u4 are immedi- 100V5 = a2~bl. ately obtainable from formulae of the type giving u 5 in the form of fifth roots, and our quest will have reached its end. Watson is using the relation zi = axi + b (i E {1, 2, 3, 4, 5}) to obtain the first equality. With z r --~ o)rul § o)2ru2 § o)3ru 3 § o)4ru 4 (r E {1, 2, 3, 4, Watson means that Ul can be given as a fifth root of an 5}) MAPLE gives e~pression involving the coefficients of the quintic, R1
and R2. ZlZ 2 § .... ZlZ 3 .... = 5(UlU4 - u2uD(~o - J - o~3 + o~4) An inspection of this pair of equations, however, suggests that we may still have a formidable task in front of us. so that
It has to be admitted that, to all intents and purposes, zlz2 + .... ZlZ3 ..... 5(ulu4 - u2u3)V~ this task is shirked by Young. In place of (8) and (9), the equations to which his analysis leads him are modified since forms of (8) and (9). They are obtainable from (8) and (9) by taking new unknowns in place of 0 and T, the new un- knowns t and 0 being given in terms of our unknowns by Consequently, to obtahi a value of 0 which satisfies the formulae Young's simultaneous equations, all that is necessary is to ob- tain a root of Cayley's sextic resolvent; and the determina- T=Ot, 02=0. tion of a rational value of ~b2 which satisfies Cayley's sextic resolvent is a perfectly straightforward process, since any Young's simultaneous equations are cubic-quartic and such value of a2~b 2 must be an integer which is a factor of quadratic-cubic respectively in ~ and t. When the original 1600000000M 2 when the coefficients in the standard form of quintic equation is Abelian, they possess a rational set of the quintic are integers, and so the number of trials which solutions. have to be made to ascertain the root is definitely limited.
Young's pair of simultaneous equations for t and 0 are The quantity M is defined on Watson's sheet 1. The con- (D 2 - ~ht2)2 + 2C(D 2 - 0t2)(C 2 ~- 30) - 8C~(D 2 + 0t 2) stant term of Cayley's sextic resolvent (0) is 40000M. + (C 2 - O)(C 2 - 5~)2 +16D02t + E2(C 2 - ~) - 2CE(D 2 - 0t 2) - 2E(C 2 - O)(C 2 + 3~) = 0 When 0 has been thus determined, Young's pair of equa- tions contain one unknown T only, and there is no diffi- and culty at all in finding the single value of T which satisfies (D + Ct)(D 2 - ~2) + t'(C 2 _ 5~)2 _ 2CDE both of them by a series of trials exactly resembling the set -Et(C 2+o)+F(C 2-~)=0. of trials by which 0 was determined.
VOLUME 24, NUMBER 4, 2002 27 Watson's method of finding a real root of the solvable where quintic equation: X = (-D + T + R1)/2, ax 5+5bx 4+ 10cx 3+10dx 2+5ex+f=0 Y=(-D-T+R2)/2, Z=-C- 0. First transform the quintic into reduced form Step 7. Determine u4 from
x 5 + 10Cx 3 § 10Dx 2 + 5Ex § F = O. UlU 4 = -C § O. Watson's step-by-step procedure gives a real root of the re- Step 8. Determine u2 from duced equation in the form x = ut + u2 + u3 + u4. The u2u2 = (-D + T - R1)/2. other four roots of the equation have the form odul + o)2Ju2 § o)3Ju3 § ~04Ju4 (j = 1, 2, 3, 4), where w = Step 9. Determine u3 from exp(2~T//5). U2U 3 = -C- O. INPUT" C,D,E,F OUTPUT" A real root of the quintic is x = Ul + u2 § Step 1. Find a POsitive integer k such that U 3 + U 4.
k]16xl0 sXM 2, The process which I have now described of solving an eV~/a is a root of (0) for e = 1 or - 1. Abelian quintic by making use of the work of both Cayley Step 2. Determine Ofrom and Young is a perfectly practical one, and, as I have al- ready implied, I have used it to solve rather more than 100 eaVk Abelian quintics. If any of you would hke to attempt the so- 0 = ION/~. lution of an Abelian quintic, you will find enough informa- Step 3. Put the value of 0 into (7) and (9) and then add tion about Ramanujan's quintic given at the foot of the third and subtract multiples of these equations as necessary sheet to enable you to complete the solution. You may re- to determine T. member that I mentioned that the equation was connected with the elliptic fimctions for which the period-quotient Step 4. Determine R1 from was X/C-~, and you will see the number 79 appearing some-
R1 = ~/(D - T) 2 § 4(C - 0)2(C § 0). what unobtrusively in the values which I have quoted for the u's. Step 5. Determine R2 from RIR2 = (C(D 2 - T 2) + (C 2 - 02)(C 2 + 302 - E))/0. This is the end of Watson's lecture. We have made a few corrections to the text: for example, in one place Watson Step 6. Determine Ul from wrote "cubic" when he clearly meant "quintic." Included = fX2y~ 1/5 in this article are the three handout sheets that he refers Ul ' to in his lecture. We conclude with three examples.
Three Examples Illustrating Watson's Procedure
Example 1. x 5 - 5x + 12 = 0 The Galois group of x 5 - 5x + 12 is D5. Here a = l, b = O, c =- O, d = O, e = - l, f =12, C=0, D=0, E=-l,F=12, K= -1, L = 3, M= -1, A = 5 X 212, ~ = 520. Equation (0) is ~b6 + 100r 4 + 6000~b2 - 256000q) - 40000 = 0. Step 1 k= 10. Step 2 1 O-X/~.
Step 3 2 T= V~. Continues on next page
28 THE MATHEMATICALINTELLIGENCER Examples (continued)
Step 4
Step 5
R2 = -2 ~/5 - ~x/-g.
Step 6 X/5 + N/5 + N/5 -V'5- ~/5- V~ 1 X = 5 ' Y = 5 , Z = --~/-~,
Ul = --( (~'5 + ~V/5 -b ~/-5)21/525 (~/'5 ~- ~/-5 -- ~f5) ) "
Step 7
u4 = - 25 ] "
Step 8 ( (~/'5 -- ~/5 -- ~/'5)2(__~'5 __ ~f5-[- -Vf5)~1/5 u2 = - 25 ] "
Step 9
u3 = - 25 ]
A solution ofx 5 - 5x + 12 = 0 is x = Ul '~ u2 ~- u3 -b u 4. This agrees with [43, Example 1].
Example 2. x 5 + 15x + 12 = 0
The Galois group of x 5 + 15x + 12 is F2o. Here a=l,b=0, c=0, d=0, e=3,f= 12, C=0, D=0, E=3, F=12, K= 3, L = 27, M = 27, A = 210 X 34, V'~ = 288V'5.
Equation (0) is ~6 _ 300r 54000r 230400X/-5& + 1080000 = 0.
Step 1 k = 180.
Step 2 3 0-~ --. 5 Step 3 T= 6_ 5" Step 4 12x/Y6 R1--- 25
Continues on next page
VOLUME 24, NUMBER 4, 2002 29 Examples (continued)
Step 5 6 ]-6 R2- 25 Step 6
15 + 6~1-6 -15 + 3X/~ 3 X- ,Y- ,Z- 25 25 5' -75 : 21v1 /1 5 ul = 125 ] Step 7
-75 + 21X//~/1/5 U 4 = 1~ ] "
Step 8
= (225 - 72X/1-0 ~1/5 U2 125 ] " Step 9 = i/ 225 +_ 72~/10 ~1/5 U3 125 ] "
This agrees with [43, Example 2].
Example 3. x 5 - 25x 3 + 50x 2 - 25 = 0
The Galois group of x 5 - 25x 3 + 50x 2 - 25 is Z/5~_. Here
a = 1, b = 0, c = -5/2, d = 5, e = O,f= -25, C = -5/2, D = 5, E = 0, F= -25, K = 75/4, L = 5375/16, M = -30625/64, h = 5 7 X 7 2 , h/~ = 5 4 x 7.
Equation (0) is
r _ 1875r 671875~b 2 _ 3500000& - 19140625 = 0. Step 1
k = 625. Step 2
0- 2 Step 3
T=O. Step 4
R1 = ~/-25 + 10X/5. Step 5
R2 = ~/-25 - 10X/5. Concludes on next page
30 THE MATHEMATICAL INTELLIGENCER Examples (continued)
Step 6 -5 + %/-25 + 10~/5 -5 + "k,/-25 - 10~5 5+~/~ X= ,Y= ,Z- -- 2 2 2 ' = {X2y~ t/5 25 + 15V'-5 + 5~v/-130 - 58X/5 u, [~-] = 4
Step 7 25 + 15X/-5 - 5%/- 130 - 5SX/5 U 4 ~-
Step 8
25 - 15~/5 + 5%/- 130 + 58~/5 U 2 =
Step 9
25 - 15%/-5 - 5X/-130 + 5SX/-5 U 3 =
REFERENCES der, The London, Edinburgh and Dublin Philosophical Magazine 1. Bruce C. Berndt, Ramanujan's Notebooks, Springer-Verlag, New and Journal of Science XXlII (1862), pp. 195, 196. [15, Vol. V, Pa- York, Part I (1985), Part II (1989), Part Ill (1991), Part IV (1994), Part per 316, p. 77.] V (1998). 10. Arthur Cayley, Final remarks on Mr. Jerrard's theory of equations 2. Bruce C. Berndt, Heng Huat Chart, and Liang-Cheng Zhang, Ra- of the fifth order, The London, Edinburgh and Dublin Philosophical manujan's class invariants and cubic continued fraction, Acta Magazine and Joumal of Science XXlV (1862), 290. [15, Vol. V, " Arithmetica 73 (1995), 67-85. Paper 321, p. 89.] 3. Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ra- 11. Arthur Cayley, Note on the solvability of equations by means of rad- manujan's class invariants, Kronecker's limit formula, and modular icals, The London, Edinburgh and Dublin Philosophical Magazine equations, Transactions of the American Mathematical Society 349 and Joumal of Science XXXVI (1868), pp. 386, 387. [15, Vol. VII, (1997), 2125-2173. Paper 421, pp. 13-14.] 4. Bruce C. Berndt, Youn-Seo Choi, and Soon-Yi Kang, The prob- 12. Arthur Cayley, On a theorem of Abel's relating to a quintic equa- lems submitted by Ramanujan to the Journal of the Indian Mathe- tion, Cambridge Philosophical Society Proceedings III (1880), matical Society, in Continued Fractions: From Analytic Number The- 155-159. [15, Vol. XI, Paper 741, pp. 132-135.] ory to Constructive Approximation, B. C. Berndt and F. Gesztesy, 13. Arthur Cayley, A solvable case of the quintic equation, Quarterly eds., Contemp. Math. No. 236, American Mathematical Society, Journal of Pure and Applied Mathematics XVIII (1882), 154-157. Providence, RI, 1999, pp. 15-56. [15, Vol. XI, Paper 777, pp. 402-404.] 5. William S. Burnside and Arthur W. Panton, The Theory of Equa- 14. Arthur Cayley, On a soluble quintic equation, American Journal of tions, 2 vols., Dover, New York, 1960. Mathematics XlII (1891 ), 53-58. [16, Vol. XIII, Paper 914, pp. 88-92.] 6. Arthur Cayley, On a new auxiliary equation in the theory of equa- 15. Arthur Cayley, The Collected Mathematical Papers of Arthur Cay- tions of the fifth order, Philosophical Transactions of the Royal So- ley, Cambridge University Press, Vol. I (1889), Vol. II (1889), Vol. Ill ciety of London CLI (1861), 263-276. [15, Vol. IV, Paper 268, pp. (1890), Vol. IV (1891), Vol. V (1892), Vol. Vl (1893), Vol. VII (1894), 309-324.] Vol. VIII (1895), Vol. IX (1896), Vol. X (1896), Vol. XI (1896), Vol. XII 7. Arthur Cayley, Note on Mr. Jerrard's researches on the equation (1897), Vol. XIIt (1897), Vol. XIV (1898). of the fifth order, The London, Edinburgh and Dublin Philosophical 16. Heng Huat Chan, Ramanujan-Weber class invariant Gn and Wat- Magazine and Journal of Science XXl (1861), 210-214. [15, Vol. son's empirical process, Journal of the London Mathematical So- V, Paper 310, pp. 50-54.] ciety 57 (1998), 545-561. 8. Arthur Cayley, On a theorem of Abel's relating to equations of the 17. James Cockle, Researches in the higher algebra, Memoirs of the Lf- fifth order, The London, Edinburgh and Dublin Philosophical Mag- erary and Philosophical Society of Manchester XV (1858), 131-142. azine and Journal of Science XXl (1861 ), 257-263. [15, Vol. V, Pa- 18. James Cockle, Sketch of a theory of transcendental roots, The per 311, pp. 55-61 .] London, Edinburgh and Dublin Philosophical Magazine and Jour- 9. Arthur Cayley, Note or1 the solution of an equation of the fifth or- nal of Science XX (1860), 145-148.
VOLUME 24, NUMBER 4, 2002 31 19. James Cockle, On the resolution of quintics, Quarterly Journal of cation to the finite algebraic solution of equations, Memoirs of the Lit- Pure and Applied Mathematics 4 (1861), 5-7. erary and Philosophical Society of Manchester XV (1859), 172-219. 20. James Cockle, Notes on the higher algebra, Quarterly Journal of 30. Robert Harley, On the theory of quintics, Quarterly Journal of Pure Pure and Applied Mathematics 4 (1861), 49-57. and Applied Mathematics 3 (1859), 343-359. 21. James Cockle, On transcendental and algebraic solution-supple- 31. Robert Harley, On the theory of the transcendental solution of al- mentary paper, The London, Edinburgh and Dublin Philosophical gebraic equations, Quarterly Journal of Pure and Applied Mathe- Magazine and Journal of Science XXlII (1862), 135-139. matics 5 (1862), 337-361. 22. Winifred A. Cooke, George Neville Watson, Mathematical Gazette 32. R. Bruce King, Beyond the Quartic Equation, BirkhAuser, Boston, 49 (1965), 256-258. 1996. 23. David A. Cox, Primes of the Form x 2 + ny2, Wiley, New York, 1989. 33. Sigeru Kobayashi and Hiroshi Nakagawa, Resolution of solvable 24. David S. Dummit, Solving solvable quintics, Mathematics of Com- quintic equation, Mathematica Japonicae ;37 (1992), 883-886. putation 57 (1991), 387-401. 34. John Emery McClintock, On the resolution of equations of the fifth 25. David S. Dummit and Richard M. Foote, Abstract Algebra, Pren- degree, American Journal of Mathematics 6 (1883-1884), 301- tice Hall, New Jersey, 1991. 315. 26. W. L. Ferrar, HigherAIgebra, Oxford University Press, Oxford, 1950. 35. John Emery McClintock, Analysis of quintic equations, American 27. Joseph A. Gallian, Contemporary Abstract Algebra, Fourth Edition, Journal of Mathematics 8 (1885), 45-84. Houghton Mifflin Co., Boston MA, 1998. 36. John Emery McClintock, Further researches in the theory of quin- 28. J. C. Glashan, Notes on the quintic, American Journal of Mathe- tic equations, American Journal of Mathematics 20 (1898), matics 7 (1885), 178-179. 157-192. 29. Robert Harley, On the method of symmetric products, and its appli- 37. Srinivasa Ramanujan, Modular equations and approximations to ~-,
32 THE MATHEMATICAL INTELLIGENCER Quarterly Journal of Mathematics 45 (1914), 350-372. [40: pp. ory: Proceedings of a Conference in Honor of Heini Halberstam, 23-39.] VoL 2, B. C. Berndt, H. G. Diamond and A. J. Hildebrand, eds., 38. Srinivasa Ramanujan, Question 699, Journal of the Indian Mathe- Birkh&user, Boston, 1996, pp. 817-838. maticalSociety 7 (1917), 199. [40: p. 331.] 62. Liang-Cheng Zhang, Ramanujan's class invariants, Kronecker's 39. Srinivasa Ramanujan, Notebooks, 2 vols., Tata Institute of Funda- limit formula and modular equations (111),Acta Arithmetica 82 (1997), mental Research, Bombay, 1957. 379-392. 40. Srinivasa Ramanujan, Collected Papers of Srinivasa Ramanujan AMS Chelsea, Providence, RI, 2000. 41. Robert A. Rankin, George Neville Watson, Journal of the London Mathematical Society 41 (1966), 551-565. 42. R. Russell, On modular equations, Proceedings of the London Mathematical Society 21 (1889-1890), 351-395. 43. Blair K. Spearman and Kenneth S. Williams, Characterization of solvable quintics x 5 + ax + b, American Mathematical Monthly 101 (1994), 986-992. 44. Blair K. Spearman and Kenneth S. Williams, DeMoivre's quintic and a theorem of Galois, Far East Journal of Mathematical Sciences 1 (1999), 137-143. 45. Blair K. Spearman and Kenneth S. Williams, Dihedral quintic poly- nomials and a theorem of Galois, Indian Journal of Pure and Ap- plied Mathematics 30 (1999), 839-845. 46. Blair K. Spearman and Kenneth S. Williams, Conditions for the in- solvability of the quintic equation x 5 + ax + b = 0, Far East Jour- nal of Mathematical Sciences 3 (2001), 209-225. 47. Blair K. Spearman and Kenneth S. Williams, Note on a paper of Kobayashi and Nakagawa, Scientiae Mathematicae Japonicae 53 (2001), 323-334. 48. K. L. Wardle, George Neville Watson, Mathematical Gazette 49 (1965), 253-256. 49. George N. Watson, Solution to Question 699, Journal of the Indian Mathematical Society 18 (1929-1930), 273-275. ~0. George N. Watson, Theorems stated by Ramanujan (XIV): a sin- gular modulus, Journal of the London Mathematical Society 6 (1931), 126-132. 51. George N. Watson, Some singular moduli (I), Quarterly Journal of Mathematics 3 (1932), 81-98. 52. George N. Watson, Some singular moduli (11), Quarterly Journal of Mathematics 3 (1932), 189-212. 53. George N. Watson, Singular moduli (3), Proceedings of the Lon- don Mathematical Society 40 (1936), 83-142. 54. George N. Watson, Singular moduli (4), Acta Arithmetica 1 (1936), 284-323. 55. George N. Watson, Singular moduli (5), Proceedings of the Lon- don Mathematical Society 42 (1937), 377-397. 56. George N. Watson, Singular moduli (6), Proceedings of the Lon- don Mathematical Society 42 (1937), 398-409. 57. Heinrich Weber, Lehrbuch derAIgebra, 3 vols., Chelsea, New York, 1961. 58. George P. Young, Resolution of solvable equations of the fifth de- gree, American Joumal of Mathematics 6 (1883-1884), 103-114. 59. George P. Young, Solution of solvable irreducible quintic equations, without the aid of a resolvent sextic, American Journal of Mathe- matics 7 (1885), 170-177. 60. George P. Young, Solvable quintic equations with commensu- rable coefficients, American Journal of Mathematics 10 (1888), 99-130. 61. Liang-Cheng Zhang, .Ramanujan's class invariants, Kronecker's limit formula and modular equations (11), in Analytic Number The-
VOLUME 24, NUMBER 4, 2002 33