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Notes Rec. R. Soc. Lond. 54 (3), 333–341 (2000) © 2000 The Royal Society
A FORGOTTEN PAPER ON THE FUNDAMENTAL THEOREM OF ALGEBRA
by
FRANK SMITHIES
167 Huntingdon Road, Cambridge CB3 0DH, UK
SUMMARY
In 1798, there appeared in the Philosophical Transactions of the Royal Society a paper by James Wood, purporting to prove the fundamental theorem of algebra, to the effect that every non-constant polynomial with real coefficients has at least one real or complex zero. Since the first generally accepted proof of this result was given by Gauss in 1799, Wood’s paper deserves careful examination. After giving a brief outline of Wood’s career, I describe the argument of his paper. His proof turns out to be incomplete as it stands, but it contains an original idea, which was to be used later, in the same context, by von Staudt, Gordan and others, without knowledge of Wood’s work. After putting Wood’s work in context, I conclude by showing how his idea can be used to prove the complex form of the fundamental theorem of algebra, stating that every non-constant polynomial with complex coefficients has at least one zero in the complex field.
INTRODUCTION
Professor Peter Goddard, Master of St John’s College, Cambridge, has drawn attention to a paper by James Wood, one of his predecessors, entitled ‘On the roots of equations’, which appeared in Philosophical Transactions volume 88. It was communicated by Nevil Maskelyne, F.R.S., the Astronomer Royal, and was read on 17 May 1798.1 The paper purports to prove what is usually called the fundamental theorem of algebra, stating it in the form ‘Every equation has as many roots of the form a±√(±b2) as it has dimensions’; in more modern language, the author is saying that a non- constant polynomial of degree n≥1, with real coefficients, has n real or complex zeros (some of which may be repeated). The paper appeared the year before Gauss, in his Helmstedt dissertation, gave the first generally accepted proof of the theorem.2 It therefore seems worthwhile to examine Wood’s paper and its context with some care.
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WOOD’S CAREER
James Wood had a remarkable career. Born on 14 December 1760, he was the son of a Lancashire weaver; he attended Bury School, and a sizarship attached to the school enabled him to enter St John’s College, Cambridge, in January 1778. As an undergraduate, he inhabited a tiny room, nicknamed the Tub (unused since his time), at the top of a staircase in Second Court; he is said to have wrapped his feet in straw to keep warm, and to have read by the light of the staircase candle. He emerged as Senior Wrangler and first Smith’s Prizeman in 1782, becoming a Fellow of the college; he became President of the college in 1802 and was elected Master in 1815. In 1816–17 he was Vice-Chancellor of the University, displaying on several occasions his attachment to older ways of doing things; in particular, he closed down the recently formed Union Society, apparently because he disapproved of any discussion of political questions by the undergraduates. He was appointed Dean of Ely in 1820, and Rector of Freshwater, Isle of Wight (a well-endowed living) in 1823. He did a great deal for St John’s College, reforming the rules for the election of Fellows and contributing substantially to the building of the college’s extension across the river (New Court). He died on 23 April 1839, leaving a considerable sum of money to the college, most of which was eventually spent in the building of a new chapel. Wood was active as a textbook writer; his Elements of Algebra3 first appeared in 1795, and went through numerous editions, continuing after his death under the editorship of Thomas Lund until 1861; it had become the standard algebra textbook in the University. It was ultimately superseded by Isaac Todhunter’s Algebra for the Use of Colleges and Schools,4 of which the first edition appeared in 1855. Wood also wrote textbooks on mechanics (1796) and optics (1798). The 1798 paper in which we are interested seems to have been Wood’s only published scientific paper, except for a note on haloes in 1790 in the Transactions of the Manchester Literary and Philosophical Society. In the Baker–Mayor history of St John’s College,5 Wood is described as a Fellow of The Royal Society, but it appears that the Society has no record of his being one.
EARLIER WORK ON THE FUNDAMENTAL THEOREM
Before we describe Wood’s paper in any detail, something should be said about the background to his work. By the time he was writing, complex numbers, generally called imaginary or impossible numbers, were widely but not universally accepted as a valid part of mathematics, especially in algebra. In a letter written by Lagrange to Lorgna in 1777,6 he remarks on the fact that the use of imaginary numbers has become generally accepted. That every algebraic equation with real coefficients has a real or imaginary root was generally believed to be true, and various attempts had been made to prove it. Those by d’Alembert, Euler, de Foncenex and Lagrange were to be severely criticized by Gauss in his 1799 dissertation.7 The principal British worker in 5. Smithies 13/9/00 11:25 AM Page 335
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this field in the 18th century was Edward Waring (1734–1798), who became Lucasian Professor of Mathematics in Cambridge in 1760. He published his Meditationes Algebraicae in 1770, and expanded it in later editions; the third edition of 17828 is quoted in Wood’s paper. In this book and in separate papers, Waring discusses various aspects of the theory of equations; in particular, he investigates several transformations to which equations can be subjected, apparently in the hope of facilitating their solution. At one point in the preface to the third edition (p. xli) he states explicitly that every equation has a real or imaginary root α+β√(–1); this remark seems to be based on a method he has devised for approximating imaginary roots. His argument is far from being a proof of the theorem. Waring clearly had good knowledge of continental work in algebra; he refers to results by Cramer, Euler, Bézout, Vandermonde, Lagrange and others. It is clear from references in Wood’s paper and in his algebra textbook that he was an admirer of Waring’s work. It is a curious fact that Wood’s paper came out only a few months before Waring’s death.
WOOD’S ARGUMENT
We shall now describe the argument used by Wood in his paper. He begins (his Prop. I) with a detailed description of the now standard procedure for calculating the highest common factor of two polynomials, the analogue for polynomials of the Euclidean algorithm for finding the highest common factor of two natural numbers. In his Prop. II, Wood starts with an equation of even degree 2m; as he remarks later, an equation of odd degree with real coefficients necessarily has a real root. He writes the equation as:
x2m +px2m–1 +qx2m–2 +rx2m–3 +…=0 (1)
and seeks two roots, which he denotes by v+z and v–z. Substituting these in (1), he obtains two equations, one of the form:
v2m +b(z)v2m–2 +c(z)v2m–4 +…=0 (2)
and the other of the form (after a factor v has been dropped):
A(z)v2m–2 +B(z)v2m–4 +C(z)v2m–6 +…=0. (3)
If we write y = v2, these become:
ym +b(z)ym–1 +c(z)ym–2 +…=0 (4) and A(z)ym–1 +B(z)ym–2 +C(z)ym–3 +…=0, (5)
the coefficients being polynomials in z. 5. Smithies 13/9/00 11:25 AM Page 336
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Applying the highest common factor (HCF) process to the left-hand sides of (4) and (5), regarded as polynomials in y, he reaches a remainder independent of y; this will be a polynomial in z of degree m(2m–1), which was later to be called the
resultant of the polynomials in (4) and (5). If now, he says, we can find a value z0 of z for which this polynomial vanishes, then the last divisor in the HCF process will be,
when z0 is substituted for z, a common factor of the two polynomials. He then alleges that the last divisor in the HCF process will be of the form y–Z(z), where Z(z)is a 2 polynomial in z, whence it would follow that v –Z(z0) is a common factor of the polynomials in (2) and (3), and that (1) has the two roots:
√ x=± (Z(z0))+z0 =z0 ±v0
√ where v0 = (Z(z0)). He then states as his conclusion (his Prop. II) that we can find a root of an equation of degree 2m, provided that we can find a root of an associated equation of degree m(2m–1). He also gives a corollary to the effect that
2 2 2 x –2z0x+z 0 –v0
will be a quadratic factor of the original equation (1). This is the first (and chief ) weak point in Wood’s argument; he gives no evidence for his statement that the last divisor in the HCF process will be of degree 1 in y; in practice, it could easily be of higher degree. We may remark here that at that period, and for a good many years afterwards, such exceptional cases were frequently ignored, attention being focused on the general case. He now states as his final conclusion (his Prop. III) that every equation (with real coefficients, it is understood) has as many roots of the form a±√(b2) as it has dimensions (i.e. its degree). His argument for this proceeds by successive enumeration of cases.
• Case 1. Every equation of odd degree has at least one real root, say x0; by division
by (x–x0), the equation can be reduced to one of lower but even degree. • Case 2. If the equation is of degree 2m, where m is odd, then m(2m–1) is odd, and therefore, by Prop. II, we have to solve an equation of odd degree, so we can find
real values for z0 and v0. The polynomial in (1) thus has a quadratic factor:
2 2 2 x –2z0 x+z0 –v0
and therefore has two real or complex zeros. • Case 3. If the equation is of degree 2m, where m is ‘evenly odd’, i.e. m/2 is odd, √ then the equation for z0 has either two real roots or two of the form a±b (–1) and 2 √ v0 will be of the form c±d (–1). He concludes that the given equation then has a biquadratic factor with real coefficients; this can be resolved into two quadratics with real coefficients, which will have four roots, as required. He goes on to say that Prop. III can be proved in the same way when m/4, m/8, m/16, 5. Smithies 13/9/00 11:25 AM Page 337
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…, is an odd number, and therefore holds for all equations. This last remark is not altogether convincing, since the situation becomes more and more complicated as the progression continues, and he gives no reason to suppose that new and unexpected complications will not arise at some stage. The argument is crying out to be recast in a form where mathematical induction can be used; as we shall see later, it is possible to do so.
LATER WORK IN THE FIELD
Although Wood’s proof is incomplete, it contains all the ingredients required for a proof of the fundamental theorem. It belongs to a class of proofs or attempted proofs that we may describe as quasi-algebraic, in that the only tool used that is not strictly algebraic is the fact that a real algebraic equation of odd degree always has a root. Later proofs belonging to this class include Gauss’s second proof of 1815,9 one given by von Staudt in 1845,10 one by Gordan in 187611 and expanded in his 1885 book12 and one by Malet in 1878.13 Wood’s paper appears to have excited no interest whatever in the mathematical world. The only mention of it that I can find, apart from a casual reference in some editions of his own algebra textbook, appears to be in Moritz Cantor’s Vorlesungen über die Geschichte der Mathematik in the final volume of 1908 in a section contributed by Florian Cajori.14 After describing Wood’s argument in some detail, he criticizes it on the same grounds that Gauss used for earlier attempts by Euler, Lagrange and Laplace: to wit, that Wood assumes that the roots exist in some sense, and that what is to be proved is that these roots must be real or complex numbers. This remark appears unjust, since Wood makes no such assumption. In further discussion, it will be useful to introduce a little terminology: if the degree of a polynomial is expressed as n=2qp, where p is odd, we shall say that the polynomial has evenness index q. The idea used by Wood of reducing an equation to one with a smaller evenness index goes back to de Foncenex’s attempted proof,15 said by Delambre to have been inspired by Lagrange, and published in 1759; it reappears in Lagrange’s paper of 1772.16 There is no evidence indicating whether Wood knew of these papers, but he seems to have been the first to use as an auxiliary equation the resultant of two polynomials derived from the original equation. This idea is also implicit in Von Staudt’s paper of 184517 and is explicitly expressed by Gordan in 1876;18 in both these papers the method is based on the same ideas as Wood’s.
THE COMPLEX FORM OF THE THEOREM
All the papers that we have mentioned so far focus on the case where the original equation has real coefficients; the fundamental theorem is thought of as being the statement that every real polynomial can be factorized into real polynomials of degree 1 or 2. The theorem that every polynomial with complex coefficients has at least one 5. Smithies 13/9/00 11:25 AM Page 338
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zero, and therefore can be factorized into complex polynomials of degree 1, goes back to 1815, when Argand outlined a proof of it;19 a more rigorous proof was given by Cauchy in 1821.20 Gauss mentioned it for the first time in his fourth proof of the theorem.21 The possibility of a quasi-algebraic proof of the complex result is hinted at by Malet22 in 1878.
A PROOF OF THE COMPLEX THEOREM BASED ON WOOD’S IDEAS
We conclude by outlining a quasi-algebraic proof of the fundamental theorem, essentially following Wood’s method, for polynomials with general real or complex coefficients. We shall take for granted some elementary properties of the resultant of two polynomials. We begin with a lemma.
Lemma. If f (x) is a polynomial of odd degree k with real or complex coefficients, then it has at least one zero. We take as our inductive hypothesis that every polynomial of odd degree less than k has a zero. If all the coefficients of f (x) are real, then consideration of the behaviour of f (x) as x varies on the real line shows at once that f (x) has at least one (real) zero. Supposing now that f (x), normalized to have leading coefficient 1, has some non- real coefficients, we write: - g(x)=f (x) f (x), - where f (x) denotes the polynomial whose coefficients are the complex conjugates of those of f(x); then g(x) has real coefficients and is of degree 2k. Write:
G(x,u)=[g(x+u)+g(x–u)]/2, H(x,u)=u–1[g(x+u)–g(x–u)]/2.
Let S(u) be the resultant of G(x,u) and H(x,u), regarded as polynomials in x. Since G(x,u) and H(x,u) are even functions of u, so is S(u); since G is of degree 2k and H of degree 2k–1, S(u) is of degree 2k(2k–1). If we write:
v=u2,
we shall have S(u)=T(v), say, where T has real coefficients and is of degree k(2k–1),
which is odd. Consequently, T has a real zero v0 and, if we write
√ u0 = v0,
we have S(u0)=0.
By known properties of the resultant, it follows that G(x,u0) and H(x,u0) have a
common divisor. Since G and H are even functions of u, and u0 is either real or 5. Smithies 13/9/00 11:25 AM Page 339
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purely imaginary, G and H will have real coefficients and therefore a real HCF, so that their common divisor h(x) may be supposed to have real coefficients and leading coefficient 1. Since:
g(x+u0)=G(x,u0)+u0 H(x,u0),
h(x) must divide g(x+u0), so h(x–u0) divides: - g(x)=f(x)f (x).
We now distinguish two cases. Suppose first that h(x–u ) is a constant multiple of f(x) - 0 or f (x), say f(x); since h(x–u0) and f(x) both have leading coefficient 1, we have
equality: h(x–u0)=f(x). In this case, h(x) and f(x) have the same degree, so h(x) has odd degree and real coefficients, and therefore has a real zero x , say. Then f(x +u ) 0 - 0 0 =h(x )=0, so f(x) has a zero x +u , as required. The case h(x–u )=f (x) can be treated 0 0 0 -0 similarly. Second, if h(x–u ) is not a constant multiple of f(x) or f (x), then some proper 0 - - divisor of h(x–u0) must divide f(x) or f (x); since the conjugate of a divisor of f (x) must divide f(x), we have a factorization f(x)=m(x) k(x), where m and k have lower degree than f. Since deg f=deg m+deg k, either deg m or deg k must be odd; by the inductive hypothesis, either m or k has a zero, which will be a zero of f, as required. We now come to the main result.
Theorem. If f(x) is an arbitrary polynomial with real or complex coefficients, then f(x) has at least one real or complex zero.
Since we have already dealt with polynomials of odd degree, we may suppose that f(x) has degree n=2qp, where p is odd and q≥1. We follow a path similar to the proof of the lemma. This time our inductive hypothesis is that: (i) every polynomial with evenness index less than q has a zero, and (ii) every polynomial with evenness index q and degree less than n has a zero. Write
F(x, u)=[f(x+u)+f(x–u)]/2, E(x, u)=u–1[f(x+u)–f(x–u)]/2.
Let R(u) be the resultant of F and E, regarded as polynomials in x; then R(u) is of degree n(n–1) and is an even function of u. Writing v=u2, we have R(u)=Q(v), say, where Q(v) is a polynomial of degree n(n–1)/2; since Q has evenness index less by 1 than that of f, it follows from part (i) of the inductive assumption that Q(v) has a zero, √ say v0, whence R(u0)=0, where u0 = v0. Consequently F(x,u0) and E(x,u0) have a common divisor, say k(x); since:
f(x+u0)=F(x,u0)+u0E(x,u0),
k(x) divides f(x+u0), so k(x–u0) divides f(x). Because deg k(x) ≤deg E(x, u0)=n–1, we 5. Smithies 13/9/00 11:25 AM Page 340
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have deg k Since f has evenness index q, either k(x–u0) or r(x) will have evenness index not greater than q, and its degree will be less than n. By part (ii) of our inductive hypothesis, either k(x–u0) or r(x) must have a zero, which will be a zero of f(x), as required. CONCLUSION We see from our discussion that, although the proof of the fundamental theorem of algebra in James Wood’s paper23 is incomplete as it stands, it can be successfully completed along his lines, and indeed completed so as to prove more than he was aiming at. In addition, he deserves the credit for being the first to see that we can construct an auxiliary polynomial of smaller evenness index than the initial one by using the resultant of two polynomials derived from the original equation, thus providing the tools for a simple quasi-algebraic proof of the fundamental theorem of algebra, even in its complex form. ACKNOWLEDGEMENTS My thanks are due to Peter Goddard for bringing Wood’s paper to my attention, to Graeme Segal and H.S. Tropp for useful discussions of Wood’s argument, and to the referee for some very helpful suggestions. NOTES 1 J. Wood, ‘On the roots of equations’, Phil. Trans. R. Soc. Lond. 88, 368–377 (1798). 2 C.G. Gauss, ‘Demonstratio nova theorematis functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse’, Dissertation, Helmstedt (1799); Werke 3, 1–30 (1866). 3 J. Wood, The elements of algebra, 9th edn (Cambridge, 1830). 4 J. Todhunter, Algebra for the use of colleges and schools (Macmillan, Cambridge, 1855). 5 T. Baker, History of the College of St John the Evangelist, Cambridge (ed. J.E.B. Mayor) (Cambridge University Press, 1869). 6 J.L. Lagrange, ‘Letter to Lorgna’, Oeuvres 14, 261. 7 Op. cit., note 2. 8 E. Waring, Meditationes Algebraicae, 3rd edn (Cambridge, 1782). 9 C.G. Gauss, ‘Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse’, Comm. Soc. Reg. Sci. Göttingen 3, 107–142 (1816); Werke 3, 33–56 (1866). 10 C. von Staudt, ‘Beweis des Satzes, dass jede algebraische rationale ganze Function von einer Veränderlichen in Factoren vom ersten Grade aufgelöst werden kann’, J. Reine Angew. Math. 29, 97–103 (1845). 11 P. Gordan, ‘Ueber den Fundamentalsatz der Algebra’, Math. Ann. 10, 572–575 (1876). 12 P. Gordan, Vorlesungen über Invariantentheorie, vol. 1 (1885). 13 J.C. Malet, ‘Proof that every algebraic equation has a root’, Trans. R. Irish Acad. 26, 453–455 (1878). 5. Smithies 13/9/00 11:25 AM Page 341 A forgotten paper on the fundamental theorem of algebra 341 14 M. Cantor (ed.), Vorlessungen über die Geschichte der Mathematik, vol. 4 (Teubner, Leipzig, 1908). 15 D. de Foncenex, ‘Réflexions sur les Quantités imaginaires’, Miscellanea Taurinensia 1, 113–146 (1759). 16 J.L. Lagrange, ‘Sur la forme des racines imaginaires des équations’, Nouv. Mém. Acad. Berlin 1772, 222–258 (1774); Oeuvres 3, 479–516 (1869). 17 Op.cit., note 10. 18 Op.cit., note 11. 19 J.R. Argand, ‘Réflexions sur la nouvelle théorie des imaginaires, suivies d’une application à la démonstration d’un théorème d’analise’, Annales de Math. (Gergonne) 5, 197–209 (1815). 20 A.L. Cauchy, Cours d’analyse de l’École Royale Polytechnique, Ie Partie. Analyse algébrique (1821); Oeuvres 2, 3 (1897). 21 C.G. Gauss, ‘Beiträge zur Theorie der algebraischen Gleichungen’, Abh. Ges. Wiss. Göttingen 4 (1848), 3–34 (1850); Werke 3, 73–102 (1866). 22 Op.cit., note 13. 23 Op.cit., note 1.