The History of Polynomials ca. 2000 BC Babylonians solve quadratics in radicals. ca. 300 BC Euclid demonstrates a geometrical construction for solving a quadratic. ca. 1000 Arab mathematicians reduce: ux2p + vxp = w to a quadratic. 1079 Omar Khayyam (1050-1123) solves cubics geometrically by intersecting parabolas and circles. ca. 1400 Al-Kashi solves special cubic equations by iteration. 1484 Nicholas Chuqet (1445?-1500?) invents a method for solving polynomials iteratively 1515 Scipione del Ferro (1465-1526) solves the cubic: x3 + mx = n but does not publish his solution. 1535 Niccolo Fontana (Tartaglia) (1500?-1557) wins a mathematical contest by solving many different cubics, and gives his method to Cardan. 1539 Girolamo Cardan (1501-1576) gives the complete solution of cubics in his book, The Great Art, or the Rules of Algebra. Complex numbers had been rejected for quadratics as absurd, but now they are needed in Cardan's formula to express real solutions. The Great Artalso includes the solution of the quartic equation by Ludovico Ferrari (1522-1565), but it is played down because it was believed to be absurd to take a quantity to the fourth power, given that there are only three dimensions. 1544 Michael Stifel (1487?-1567) condenses the previous eight formulas for the roots of a quadratic into one. 1593 Francois Viete (1540-1603) solves the casus irreducibilisof the cubic using trigonometric functions. 1594 Viete solves a particular 45th degree polynomial equation by decomposing it into cubics and a quintic. Later he gives a solution of the general cubic that needs the extraction of only a single cube root. 1629 Albert Girard (1595-1632) conjectures that the nthdegree equation has n roots counting multiplicity. 1637 Rene Descartes (1596-1650) gives his rule of signs to determine the number of positive roots of a given polynomial 1666 Isaac Newton (1642-1727) finds a recursive way of expressing the sum of the roots to a given power in terms of the coefficients. 1669 Newton introduces his iterative method for the numerical approximation of roots. 1676 Newton invents Newton's parallelogram to approximate all the possible values of y in terms of x, if: Sigma(i, j = 0 -> n) [aij xiyj] = 0 1683 Ehrenfried Walther von Tschirnhaus (1646-1716) generalizes the linear substitution that eliminates the xn-1term in the nth degree polynomial to eliminate the xn-2and xn-3 terms as well. Gottfried Wilhelm Leibniz (1646-1716) had pointed out that trying to get rid of the xn-4term usually leads to a harder equation than the original one. 1691 Michael Rolle (1652-1719) proves that f'(x) has an odd number of roots in the interval between two successive roots of f(x). 1694 Edmund Halley (1656-1742) discusses interative solutions of quartics with symbolic coefficients 1728 Daniel Bernoulli (1700-1782) expresses the largest root of a polynomial as the limit of the ratio of the successive power sums of the roots. 1732 Leonard Euler (1707-1783) tries to find solutions of polynomial equations of degree n as sums of nth roots, but fails. 1733 Halley solves the quadratic in trigonometric functions. 1748 Colin Maclaurin (1698-1746) generalizes Newton's relations for powers greater than the degree of the polynomial. 1757 Johann Heinrich Lambert (1728-1777) gives series solutions of trinomial equations: xp + x + r = 0 1762 Etienne Bezout (1730-1783) tries to find solutions of polynomial equations of degree n as linear combinations of powers of an nth root of unity, but fails. 1762 Euler tries to find solutions of polynomial equations of degree n as linear combinations of powers of an nth root, but fails. 1767 Joseph Louis Lagrange (1736-1813) expresses the real roots of a polynomial equation in terms of a continued fraction. 1769 Lagrange expands a function as a series in powers of another function and uses this to solve trinomial equations. 1770 Lagrange shows that polynomials of degree five or more cannot be solved by the methods used for quadratics, cubics, and quartics. He introduces the Lagrangeresolvent, an equation of degree n!. 1770 Euler gives series solutions of: xm+n + axm + bxn = 0 1770 John Rowning (1699-1771) develops the first mechanical device for solving polynomial equations. Although the machine works for any degree in theory, it was only practical for quadratics. 1771 Gianfrancesco Malfatti (1731-1807), starting with a quintic, finds a sextic that factors if the quintic is solvable in radicals. 1772 Lagrange finds a stationary solution of the three body problem that requires the solution of a quintic. 1786 Erland Samuel Bring (1736-1798) proves that every quintic can be transformed to: z5 + az + b = 0 1796 Jean Baptiste Joseph Fourier (1768-1830) determines the maximum number of roots in an interval. 1799 Paolo Ruffini (1765-1822) publishes the book, General Theory of Equations, in which the Algebraic Solution of General Equations of a Degree Higher than the Fourth is Shown to Be Impossible 1799 Carl Friedrich Gauss (1777-1855) proves the fundamental theorem of algebra: Every nonconstant polynomial equation has at least one root. 1801 Gauss solves the cyclotomic equation: z17 = 1 in square roots. 1819 William George Horner (1768-1847) presents his rule for the efficient numerical evaluation of a polynomial. Ruffini had proposed a similar idea. 1826 WilNiels Henrik Abel (1802-1829) publishes Proof of the Impossibility of Generally Solving Algebraic Equations of a Degree Higher than the Fourth. 1829 Jacques Charles Francois Sturm (1803-1855) finds the number of real roots of a given polynomial in a given interval. 1829 Carl Gustav Jacobi (1804-1851) studies modular equations for elliptic functions which are fundamental for Hermite's 1858 solution of quintics. 1831 Augistin-Louis Cauchy (1789-1857) determines how many roots of a polynomial lie inside a given contour in the complex plane. 1832 Evariste Galois (1811-1832) writes down the main ideas of his theory in a letter to Auguste Chevalier the day before he dies in a duel 1832 Friedrich Julius Richelot (1808-1875) solves the cycolotomic equation: z257 = 1 in square roots. 1834 George Birch Jerrard (1804-1863) shows that every quintic can be transformed to: z5 + az + b = 0 1837 Karl Heinrich Graeffe (1799-1873) invents a widely used method to determine numerical roots by hand. Similar ideas had already been suggested independently by Edward Waring (1734-1798), Germinal Pierre Dandelin (1794-1847), Moritz Abraham Stern (1807-1894), and Nickolai Lobachevski (1792-1856). Johann Franz Encke (1791-1865) later perfects the method. 1838 Pafnuti Chebyshev (1821-1894) generalizes Newton's method to make the convergence arbitrarily fast and uses this to approximate the roots of polynomials. 1840 L. Lalanne builds a practical machine to solve polynomials up to degree seven. 1844 Gotthold Eisenstein (1823-1852) gives the first few terms of a series for one root of a canonical quintic. 1854 Josef Ludwig Raabe (1801-1859) transforms the problem of finding roots to solving a partial differential equation, obtaining explicit roots for a quadratic. 1858 Charles Hermite (1822-1901), Leopold Kronecker (1823-1891), and Francesco Brioschi (18241897) independently solve a quintic in Bring-Jerrard form explicitly in terms of elliptic modular functions. 1860, 1862 James Cockle (1819-1895) and Robert Harley (1828-1910) link a polynomial's roots to differential equations. 1861 Carl Johan Hill (1793-1863) remarks that Jerrard's 1834 work is contained in Bring's 1786 work. 1862 William Hamilton (1805-1865) closes some gaps in Abel's impossibility proof. 1869 Johannes Karl Thomae (1840-1921) discovers a key ingredient for the representation of roots using Siegel functions. 1870 Camille Jordan (1838-1922) shows that algebraic equations of any degree can be solved in terms of modular functions. 1871 Ludwig Sylow (1832-1918) puts the finishing touches onGalois's proofs on solvability. 1873 Hermann Amandus Schwarz (1843-1921) investigates the relationship between hypergeometric differential equations and the group structure of the Platonic solids, an important part of Klein's solution to the quintic. 1877 Felix Klein (1849-1925) solves the icosahedral equation in terms of hypergeometric functions. This allows him to give a closed-form solution of a principal quintic. 1884, 1892 Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of theta functions. 1885 John Stuart Cadenhead Glashan (1844-1932), George Paxton Young (1819-1889), and Carl Runge (1856-1927), show that all irreducible solvable quintics with the quadratic, cubic, and quartic terms missing have a spezial form. 1890, 1891 Vincenzo Mollame (1848-1912) and Ludwig Otto Hoelder (1859-1937) prove the impossibility of avoiding intermediate complex numbers in expressing the three roots of a cubic when they are all real. 1891 Karl Weierstrass (1815-1897) presents an interation scheme that simultaneously determines all the roots of a polynomial. 1892 David Hilbert (1862-1943) proves that for every n there exists an nth polynomial with rational coefficients whoseGalois group is the symmetric group Sn 1894 Johann Gustav Hermes (1846-1912) completes his 12-year effort to calculate the 65537th root of unity using square roots. 1895 Emory McClintock (1840-1916) gives series solutions for all the roots of a polynomial. 1895-1910 Klein, Leonid Lachtin (1858-1927), Paul Gordan (1837-1912), Heinrich Maschke (1853-1908), Arthur Byron Coble (1878-1966), Frank Nelson Cole (1861-1926), and Anders Wiman (18651959) develop the fundamentals of how to solve a sextic via Klein's approach. 1915 Robert Hjalmal Mellin (1854-1933) solves an arbitrary polynomial equation with Mellin integrals. 1905-1925 R. Birkeland shows that the roots of an algebraic equation can be expressed using hypergeometric functions in several variables. Alfred Capelli (1855-1910), Guiseppe Belardinelli (1894-?), and Salvatore Pincherle (1853-1936) express related ideas. 1926 Paul Emile Appell (1855-1930) and Joseph Marie Kampe de Feriet (1893-1982) recognize the hypergeometric functions in the series solution of the quintic. 1932 Andre Bloch (1893-1948) and George Polya (1887-1985) investigate the zeros of polynomials of arbitrary degree with random coefficients. 1934 Richard Brauer (1901-1977) analyzes Klein's solution of the quintic using the theory of fields. 1937 Scientists at Bell Labs build the Isograph, a precision instrument that calculates roots of polynomials up to degree 15. 1938, 1942 Emil Artin (1898-1962) uses field theory to develop the modern theory of algebraic equations. 1957 Vladimir Arnol'd, using results of Andrei Kolmogorov (1903-1987), shows that it is possible to express the roots of the reduced 7th degree polynomial in continuous functions of two variables, answering Hilbert's 13th problem in the negative. 1984 Hiroshi Umemura expresses the roots of an arbitrary polynomial through elliptic Siegel functions. 1989 Peter Doyle and Curt McMullen construct a generally convergent, purely iterative algorithm for the numerical solution of a reduced quintic, relying on the icosahedral equation. 1991, 1992 David Dummit and (independently) Sigeru Kobayashi and Hiroshi Nakagawa give methods for finding the roots of a general solvable quintic in radicals. History of the Quartic In 1540, Cardan was given the following problem: Divide 10 into 3 parts: The parts are in continued proportion and the product of the first 2 is 6 This problem lead to a quartic which Cardan was not able to solve. He gave it to Ferrari. Ferrari was the first to develop an algebraic technique for solving the general quartic. He applied his technique (which was published by Cardano ) to the equation x4 + 6x2 - 60x + 36 = 0