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A Forgotten Paper on the Fundamental Theorem of Algebra 5. Smithies 13/9/00 11:25 AM Page 333 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. 333 5. Smithies 13/9/00 11:25 AM Page 334 334 Frank Smithies 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 A forgotten paper on the fundamental theorem of algebra 335 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 336 336 Frank Smithies 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.
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