Index
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:
2p p
ux + vx = w
to a quadratic.
1079
 Omar Khayyam (10501123) solves cubics geometrically by
intersecting parabolas and circles.
ca. 1400
 AlKashi solves special cubic equations by iteration.
1484
 Nicholas Chuqet (1445?1500?) invents a method for solving
polynomials iteratively.
1515
 Scipione del Ferro (14651526) solves the cubic:
3
x + mx = n
but does not publish his solution.
1535
1539
 Girolamo Cardan (15011576) 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
Art also includes the solution of the quartic equation by
Ludovico Ferrari (15221565), 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 (15401603) solves the casus irreducibilis
of the cubic using trigonometric functions.
1594
 Viete solves a particular
45 th 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 (15951632) conjectures that the
n th
degree equation has n roots counting multiplicity.
1637
 Rene Descartes (15961650) gives his rule of signs to determine the
number of positive roots of a given polynomial.
1666
 Isaac Newton (16421727) 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
1683
 Ehrenfried Walther von Tschirnhaus
(16461716) generalizes the linear
substitution that eliminates the
x^(n1) term in the
n th degree polynomial to eliminate the
x^(n2) and x^(n3) terms as well. Gottfried
Wilhelm Leibniz (16461716) had pointed out that trying to get rid
of the x^(n4) term usually leads to a harder equation
than the original one.
1691
 Michael Rolle (16521719) proves that
f'(x) has an odd
number of roots in the interval between two successive roots of
f(x) .
1694
 Edmund Halley (16561742) discusses interative solutions of quartics
with symbolic coefficients.
1728
 Daniel Bernoulli (17001782) expresses the largest root of a polynomial
as the limit of the ratio of the successive power sums of the roots.
1732
 Leonard Euler (17071783) tries to
find solutions of polynomial
equations of degree
n as sums of n th
roots, but fails.
1733
 Halley solves the quadratic in trigonometric functions.
1748
 Colin Maclaurin (16981746) generalizes Newton's relations for powers
greater than the degree of the polynomial.
1757
1762
 Etienne Bezout (17301783) tries to find solutions of polynomial
equations of degree
n as linear combinations of powers
of an n th root of unity, but fails.
 Euler tries to find solutions of polynomial equations of degree
n as linear combinations of powers of an
n th root, but fails.
1767
 Joseph Louis Lagrange (17361813)
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 Lagrange resolvent, an equation of degree
n! .
 Euler gives series solutions of:
m+n m n
x + ax + bx = 0
 John Rowning (16991771) 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
1772
 Lagrange finds a stationary solution of the three body problem that
requires the solution of a quintic.
1786
 Erland Samuel Bring (17361798) proves that every quintic can be
transformed to:
5
z + az + b = 0
1796
 Jean Baptiste Joseph Fourier (17681830) determines the maximum
number of roots in an interval.
1799
 Paolo Ruffini (17651822) 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.
 Carl Friedrich Gauss (17771855)
proves the fundamental theorem
of algebra: Every nonconstant polynomial equation has at least
one root.
1801
 Gauss solves the cyclotomic equation:
17
z = 1
in square roots.
1819
 William George Horner (17681847) presents his rule for the efficient
numerical evaluation of a polynomial. Ruffini had proposed a similar
idea.
1826
 Niels Henrik Abel (18021829)
publishes Proof of the Impossibility
of Generally Solving Algebraic Equations of a Degree Higher than the
Fourth.
1829
1831
 AugistinLouis Cauchy (17891857) determines how many roots of a
polynomial lie inside a given contour in the complex plane.
1832
 Evariste Galois (18111832) writes
down the main ideas of his theory
in a letter to Auguste Chevalier the day before he dies in a duel.
 Friedrich Julius Richelot (18081875) solves the cycolotomic equation:
257
z = 1
in square roots.
1834
 George Birch Jerrard (18041863) shows that every quintic can be
transformed to:
5
z + az + b = 0
1837
 Karl Heinrich Graeffe (17991873) invents a widely used method to
determine numerical roots by hand. Similar ideas had already been
suggested independently by Edward Waring (17341798), Germinal
Pierre Dandelin (17941847), Moritz Abraham Stern (18071894),
and Nickolai Lobachevski (17921856). Johann Franz Encke
(17911865) later perfects the method.
1838
 Pafnuti Chebyshev (18211894)
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 (18231852)
gives the first few terms of a
series for one root of a canonical quintic.
1854
 Josef Ludwig Raabe (18011859) transforms the problem of finding
roots to solving a partial differential equation, obtaining
explicit roots for a quadratic.
1858
1860, 1862
 James Cockle (18191895) and
Robert Harley (18281910) link a
polynomial's roots to differential equations.
1861
 Carl Johan Hill (17931863) remarks that Jerrard's 1834 work is
contained in Bring's 1786 work.
1862
 William Hamilton (18051865) closes some gaps in Abel's impossibility
proof.
1869
 Johannes Karl Thomae (18401921) discovers a key ingredient for the
representation of roots using Siegel functions.
1870
 Camille Jordan (18381922) shows that algebraic equations of any degree
can be solved in terms of modular functions.
1871
 Ludwig Sylow (18321918) puts the finishing touches on Galois's proofs
on solvability.
1873
 Hermann Amandus Schwarz (18431921) 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 (18491925) solves
the icosahedral equation in terms of
hypergeometric functions. This allows him to give a closedform
solution of a principal quintic.
1880s
 Tables, nomograms, and various mechanical devices are constructed
for solving trinomial equations.
1884, 1892
 Ferdinand von Lindemann (18521939) expresses the roots of an
arbitrary polynomial in terms of theta functions.
1885
1887
 Carl Woldemar Heymann (18851910) solves trinomial equations using
integrals.
1890, 1891
 Vincenzo Mollame (18481912) and Ludwig Otto Hoelder (18591937)
prove the impossibility of avoiding intermediate complex numbers
in expressing the three roots of a cubic when they are all real.
1891
 Karl Weierstrass (18151897) presents an interation scheme that
simultaneously determines all the roots of a polynomial.
1892
 David Hilbert (18621943) proves that for every
n
there exists an n th polynomial with
rational coefficients whose Galois group is the symmetric
group Sn . (The same is true for the alternating
group An .)
1894
 Johann Gustav Hermes (18461912) completes his 12year effort to
calculate the
65537 th root of unity using
square roots.
1895
 Emory McClintock (18401916) gives series solutions for all the
roots of a polynomial.
 Leonardo Torres Quevedo (18521936) builds a machine for the
mechanical calculation of the real and complex roots of an
arbitrary trinomial equation.
18951910
 Klein, Leonid Lachtin (18581927), Paul Gordan (18371912),
Heinrich Maschke (18531908), Arthur Byron Coble (18781966),
Frank Nelson Cole (18611926), and Anders Wiman (18651959)
develop the fundamentals of how to solve a sextic via Klein's
approach.
1915
 Robert Hjalmal Mellin (18541933) solves an arbitrary polynomial
equation with Mellin integrals.
19051925
 R. Birkeland shows that the roots of an algebraic equation can be
expressed using hypergeometric functions in several variables.
Alfred Capelli (18551910), Guiseppe Belardinelli (1894?), and
Salvatore Pincherle (18531936) express related ideas.
1926
1932
 Andre Bloch (18931948) and George Polya (18871985) investigate the zeros of
polynomials of arbitrary degree with random coefficients.
1934
 Richard Brauer (19011977) 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 (18981962) uses field theory to develop the modern
theory of algebraic equations.
1957
 Vladimir Arnol'd, using results of Andrei Kolmogorov (19031987),
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.
   
 
