- THEORY OF EQUATIONS 441
Euler. In his 1749 paper "Recherches sur les racines imaginaires des equations"
("Investigations into the imaginary roots of equations"), devoted to equations whose
degree is a power of 2 and published in the memoirs of the Berlin Academy, Euler
showed that when the coefficients of a polynomial are real, its roots occur in con-
jugate pairs, and therefore produce irreducible real quadratic factors of the form
(x - a)^2 + b^2. In this paper Euler argued that every polynomial of degree 2n with
real coefficients can be factored as a product of two polynomials of degree 2"_1
with real coefficients. In the course of the proof Euler presented the germ of an
idea that was to have profound consequences. In showing that a polynomial of
degree 8 could be written as a product of two polynomials of degree 4, he assumed
that the coefficient of x^7 was made equal to zero by means of a linear substitution.
The remaining polynomial ÷^8 - ax^6 + bx^5 - cx* - dx^2 + ex — f was then to be
written as a product(x^4 - ux^3 + ax^2 + â÷ + 7)(x^4 + ux^3 + ä÷^2 + å÷ + æ).
Euler noted that since u was the sum of four roots of the equation, it could assume
(potentially) 70 values (the number of combinations of eight things taken four at a
time), and its square would satisfy an equation of degree 35.
In this paper, Euler also conjectured that the roots of an equation of degree
higher than 4 cannot be constructed by applying a finite number of algebraic oper-
ations to the coefficients. This was the first explicit statement of such a conjecture.
In his 1762 paper "De resolutione aequationum cuiusque gradus" ("On the
solution of equations of any degree"), published in the proceedings of the Petersburg
Academy, Euler tried a different approach,^3 assuming a solution of the formx = w + AS/v + B y/÷â + •·· + £? vV^1 "^1 ,
where w is a real number and í and the coefficients A,...,Q are to be found by a
procedure resembling a Tschirnhaus transformation. This approach was useful for
equations of degree 2", but fell short of being a general solution of the problem.D'Alembert. Euler's contemporary and correspondent Jean le Rond d'Alembert
(1717-1783) tried to prove that all polynomials could be factored into linear and
quadratic factors in order to prove that all rational functions could be integrated
by partial fractions. In the course of his argument he assumed that any algebraic
function could be expanded in a scries of fractional powers of the independent
variable. While Euler was convinced by this proof, he also wrote to d'Alembert to
say that this assumption would be questioned (Bottazzini, 1986, pp. 1518).
Lagrange. In 1770 Lagrange made a survey of the methods known up to his time
for solving general equations. He devoted a great deal of space to a preliminary
analysis of the cubic and quartic equations. In particular, he was intrigued by the
fact that the resolvent equation, which he called the reduced equation (equation
reduite), for the cubic was actually an equation of degree 6 that just happened to
be quadratic in the third power of the unknown. He showed that if the roots of
the cubic equation x^3 + px = q being solved were o, 6, and c, then a root of the
resolvent would be
a + ab + a^2 c
(^3) This approach was discovered independently by Etienne Bezout (1730 1783).