An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 89. 90.

Rtoots of' algebraic equations.

153

vanishes; some of these vanishing points may coincide, but always the
total sum of their nullitudes is n. This theorem, which can also be
stated in the words: Every equation of the n't degree
f(z) = a0 + a1 + a2 + * a,, n = 0
has n complex roots; is known as the Fundamental Theorem of
Algebra.*) When the coefficients a, al... a n are all real, if the
equation have one complex root z = a +- iP, it has also the conjugate
complex root - a -- ii; for then we have in general:
-(a -- i ) - i V, if f(a + i ) - + i V.
But, when a +- if is a root, both U and V vanish.
The implicit algebraic function, defined by the equation ~ 25:
Ao + A9- + * *y2 Ayn =- 0,
in which A0... A,, signify integer polynomials in x, is accordingly,
when complex solutions also are taken into account, an n-valued function,
i. e. to each value of x, for which the values of the coefficients A0... A
are determinate, belong n equal or different values of y, namely the
n roots of this equation of the Wth degree.
The calculation of the n, values of the roots, i. e. their expression
as functions of the coefficients, forms the object of the Theory of
Equations. As long as n <4, the roots can be developed in a closed
form as functions of the coefficients by help of the explicit algebraical
operations of the first six species; if n > 4 the solution of the general
equation gives rise to new functions whose properties we have to
investigate in the next Chapter. But it is in all cases possible when
the coefficients of an equation are given in the form of determinate
numerical quantities, to express each root numerically with any
required degree of approximation, i.e. after the method of inclusion
within limits to form two infinite series of rational numbers, whereof
one has the real constituent, and the other the factor of the imaginary
constituent of a root as its limiting value.
*) The theorem was first proved by Gauss in his doctoral dissertation 1799,
to this there is a supplement of the year 1849; Gauss published two other proofs in
1815 and 1816 (Werke, Vol. III). The proof detailed in the text, applicable as
it is to infinite series of powers, is derived essentially from Cauchy (Journal
de l'Ecole polytechnique, Cahier 25, 1837); he had previously, in his Analyse
alg6brique, chap. X, 1821, given an elementary proof for the existence of the in
roots of an equation, that coincides in principle with that developed by Argand
(Gergonne Ann., Vol. V, 1815).