RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

58 Chapter 5. Towards unsolvable equations
polynomial equations were complex, i.e. of the form a + b √ −1 for real a, b, was raised;
and around the time of C. F. GAUSS (1777–1855), the theorem acquired the name of
the Fundamental Theorem of Algebra.
When G. W. LEIBNIZ (1646–1716) doubted that the polynomial x 4 + c 4 could be
split into two real factors of the second degree, 5 the validity of the result seemed for a
moment in doubt. L. EULER (1707–1783) demonstrated in 1749 (published 1751) that
the set of complex numbers was closed under all algebraic and numerous transcendental
operations. 6 Thus, at least by 1751 it would implicitly be known that √ i = 1+i
√ 2 .
This made LEIBNIZ’S supposed counter-example evaporate, since he factorized his
polynomial as
x 4 + c 4 �
= x 2 �
− ic
2�
x 2 + ic 2�
�
= x − √ � �
ic x + √ � �
ic x + √ � �
−ic x − √ �
−ic .
Numerous prominent mathematicians of the eighteenth century — among them no-
tably J. LE R. D’ALEMBERT (1717–1783), EULER, and J. L. LAGRANGE (1736–1813) —
sought to provide proofs that any real polynomial could be split into linear and quadratic
factors which would prove that any imagined roots were indeed complex. In the half-
century 1799–1849 GAUSS gave a total of four proofs 7 which, although belonging to an
emerging trend of indirect existence proofs, were considered to be superior in rigour
when compared to those of his predecessors.
Characterizing roots. The proofs of the Fundamental Theorem of Algebra were mostly
existence proofs which did not provide any information on the computational aspect.
Other similar, nonconstructive results were also pursued. An important subfield of the
theory of equations was developed in order to characterize and describe properties of
the roots of a given equation from a priori inspections of the equation and without
explicitly knowing the roots.
LAGRANGE’S study of the properties of the roots of particular equations was an
offspring from his attempts to solve higher degree equations through algebraic ex-
pressions (see below). 8 LAGRANGE’S interest in numerical equations, i.e. concrete
equations in which some dependencies among the coefficients can exist, can be di-
vided into three topics: the nature and number of the roots, limits for the values of
these roots, and methods for approximating these. LAGRANGE made use of analytic
geometry, function theory, and the Lagrangian calculus in order to investigate these
topics. 9
5 (K. Andersen, 1999, 69).
6 (ibid., 70).
7 GAUSS’ proofs can be found in (C. F. Gauss, 1863–1933, vol. 3) and have been collected in German
translation in (C. F. Gauss, 1890).
8 (Hamburg, 1976, 28).
9 (ibid., 29–30).