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The History of Polynomials ca. 2000 BCBabylonians solve quadratics in radicals.ca. 300 BCEuclid?demonstrates a geometrical construction for solving a quadratic.ca. 1000Arab mathematicians reduce:ux2p?+ vxp?= wto a quadratic.1079Omar?Khayyam?(1050-1123) solves cubics geometrically by intersecting parabolas and circles.ca. 1400Al-Kashi solves special cubic equations by iteration.1484Nicholas Chuqet (1445?-1500?) invents a method for solving polynomials iteratively1515Scipione del Ferro (1465-1526) solves the cubic:x3?+ mx = nbut does not publish his solution.1535Niccolo Fontana (Tartaglia) (1500?-1557) wins a mathematical contest by solving many different cubics, and gives his method to Cardan.1539Girolamo 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.1544Michael Stifel (1487?-1567) condenses the previous eight formulas for the roots of a quadratic into one.1593Francois?Viete?(1540-1603) solves the?casus irreducibilisof the cubic using trigonometric functions.1594Viete?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.1629Albert Girard (1595-1632) conjectures that the?nthdegree equation has?n?roots counting multiplicity.1637Rene?Descartes?(1596-1650) gives his rule of signs to determine the number of positive roots of a given polynomial1666Isaac?Newton?(1642-1727) finds a recursive way of expressing the sum of the roots to a given power in terms of the coefficients.1669Newton?introduces his iterative method for the numerical approximation of roots.1676Newton?invents?Newton's parallelogram to approximate all the possible values of?y?in terms of?x, if:Sigma(i, j = 0 -> n) [aij?xiyj] = 01683Ehrenfried 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.1691Michael?Rolle?(1652-1719) proves that?f'(x)?has an odd number of roots in the interval between two successive roots of?f(x).1694Edmund Halley (1656-1742) discusses interative solutions of quartics with symbolic coefficients1728Daniel Bernoulli (1700-1782) expresses the largest root of a polynomial as the limit of the ratio of the successive power sums of the roots.1732Leonard?Euler?(1707-1783) tries to find solutions of polynomial equations of degree?n?as sums of?nth?roots, but fails.1733Halley solves the quadratic in trigonometric functions.1748Colin Maclaurin (1698-1746) generalizes?Newton's relations for powers greater than the degree of the polynomial.1757Johann Heinrich Lambert (1728-1777) gives series solutions of trinomial equations:xp?+ x + r = 01762Etienne 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.1762Euler?tries to find solutions of polynomial equations of degree?n?as linear combinations of powers of an?nth?root, but fails.1767Joseph Louis?Lagrange?(1736-1813) expresses the real roots of a polynomial equation in terms of a continued fraction.1769Lagrange?expands a function as a series in powers of another function and uses this to solve trinomial equations.1770Lagrange?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!.1770Euler?gives series solutions of:xm+n?+ axm?+ bxn?= 01770John 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.1771Gianfrancesco Malfatti (1731-1807), starting with a quintic, finds a sextic that factors if the quintic is solvable in radicals.1772Lagrange?finds a stationary solution of the three body problem that requires the solution of a quintic.1786Erland Samuel Bring (1736-1798) proves that every quintic can be transformed to:z5?+ az + b = 01796Jean Baptiste Joseph?Fourier?(1768-1830) determines the maximum number of roots in an interval.1799Paolo 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 Impossible1799Carl Friedrich?Gauss?(1777-1855) proves the fundamental theorem of algebra: Every nonconstant polynomial equation has at least one root.1801Gauss?solves the cyclotomic equation:z17?= 1in square roots.1819William George Horner (1768-1847) presents his rule for the efficient numerical evaluation of a polynomial. Ruffini had proposed a similar idea.1826WilNiels Henrik?Abel?(1802-1829) publishes?Proof of the Impossibility of Generally Solving Algebraic Equations of a Degree Higher than the Fourth.1829Jacques Charles Francois Sturm (1803-1855) finds the number of real roots of a given polynomial in a given interval.1829Carl Gustav?Jacobi?(1804-1851) studies modular equations for elliptic functions which are fundamental for Hermite's 1858 solution of quintics.1831Augistin-Louis?Cauchy?(1789-1857) determines how many roots of a polynomial lie inside a given contour in the complex plane.1832Evariste?Galois?(1811-1832) writes down the main ideas of his theory in a letter to Auguste Chevalier the day before he dies in a duel1832Friedrich Julius Richelot (1808-1875) solves the cycolotomic equation:z257?= 1in square roots.1834George Birch Jerrard (1804-1863) shows that every quintic can be transformed to:z5?+ az + b = 01837Karl 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.1838Pafnuti Chebyshev (1821-1894) generalizes?Newton's method to make the convergence arbitrarily fast and uses this to approximate the roots of polynomials.1840L. Lalanne builds a practical machine to solve polynomials up to degree seven.1844Gotthold Eisenstein (1823-1852) gives the first few terms of a series for one root of a canonical quintic.1854Josef Ludwig Raabe (1801-1859) transforms the problem of finding roots to solving a partial differential equation, obtaining explicit roots for a quadratic.1858Charles Hermite (1822-1901), Leopold Kronecker (1823-1891), and Francesco Brioschi (1824-1897) independently solve a quintic in Bring-Jerrard form explicitly in terms of elliptic modular functions.1860, 1862James Cockle (1819-1895) and Robert Harley (1828-1910) link a polynomial's roots to differential equations.1861Carl Johan Hill (1793-1863) remarks that Jerrard's 1834 work is contained in Bring's 1786 work.1862William Hamilton (1805-1865) closes some gaps in?Abel's impossibility proof.1869Johannes Karl Thomae (1840-1921) discovers a key ingredient for the representation of roots using Siegel functions.1870Camille Jordan (1838-1922) shows that algebraic equations of any degree can be solved in terms of modular functions.1871Ludwig Sylow (1832-1918) puts the finishing touches onGalois's proofs on solvability.1873Hermann 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.1877Felix 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, 1892Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of theta functions.1885John 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, 1891Vincenzo 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.1891Karl Weierstrass (1815-1897) presents an interation scheme that simultaneously determines all the roots of a polynomial.1892David?Hilbert?(1862-1943) proves that for every?n?there exists an?nth?polynomial with rational coefficients whoseGalois?group is the symmetric group?Sn1894Johann Gustav Hermes (1846-1912) completes his 12-year effort to calculate the?65537th?root of unity using square roots.1895Emory McClintock (1840-1916) gives series solutions for all the roots of a polynomial.1895-1910Klein, 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 (1865-1959) develop the fundamentals of how to solve a sextic via Klein's approach.1915Robert Hjalmal Mellin (1854-1933) solves an arbitrary polynomial equation with Mellin integrals.1905-1925R. 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.1926Paul Emile Appell (1855-1930) and Joseph Marie Kampe de Feriet (1893-1982) recognize the hypergeometric functions in the series solution of the quintic.1932Andre Bloch (1893-1948) and George Polya (1887-1985) investigate the zeros of polynomials of arbitrary degree with random coefficients.1934Richard Brauer (1901-1977) analyzes Klein's solution of the quintic using the theory of fields.1937Scientists at Bell Labs build the Isograph, a precision instrument that calculates roots of polynomials up to degree 15.1938, 1942Emil Artin (1898-1962) uses field theory to develop the modern theory of algebraic equations.1957Vladimir 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.1984Hiroshi Umemura expresses the roots of an arbitrary polynomial through elliptic Siegel functions.1989Peter 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, 1992David 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 6This 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 equationx4?+ 6x2?- 60x + 36 = 0 ................
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