- THEORY OF EQUATIONS 443
The technique of counting the number of different values the root of the re-
solvent will have when the roots of the original equation are permuted among
themselves was an important clue in solving the problem of the quintic.
1.5. Gauss and the fundamental theorem of algebra. The question of the
theoretical existence of roots was settled on an intuitive level in the 1799 dissertation
of Gauss. Gauss distinguished between the abstract existence of a root, which he
proved, and an algebraic algorithm for finding it, the existence of which he doubted.
He pointed out that attempts to prove the existence of a root and any possible
algorithm for finding it must assume the possibility of extracting the nth root of a
complex number. He also noted the opinion, first stated by Euler, that no algebraic
algorithm existed for solving the general quintic.
The reason we say that the existence of roots was settled only on the intuitive
level is that the proof of the fundamental theorem of algebra is as much topological
as algebraic. The existence of real roots of an equation of odd degree with real
coefficients seems obvious since a real polynomial of odd degree tends to oppositely
signed infinities as the independent variable ranges from one infinity to the other.
It thus follows by connectivity that it must assume a zero at some point. Gauss'
proof of the existence of complex roots was similar. Much of what he was doing was
new at the time, and he had to explain it in considerable detail. For that reason, he
preferred to use only real-variable methods, so as not to raise any additional doubts
with the use of complex numbers. In fact, he stated his purpose in that way: to
prove that every equation with real coefficients has a complete factorization into
linear and quadratic real polynomials.
The complex-variable background of the proof is obvious nowadays, and Gauss
admitted that his lemmas were normally proved using complex numbers. The steps
were as follows. First, considering the equation zm + Azm~l + Bzm~^2 + • •• +
Kz^2 + Lz + Ì = 0, where all coefficients A,..., Ì were real numbers,^4 taking
æ = r(cos</? + isinip) and using the relation zm — rm (cos ôçö + i sin ôçö), one can
see that finding a root amounts to setting the real and imaginary parts equal to
zero simultaneously, that is, finding r and ö such that
rm cosnup + Arm~l cos(m — \)ø + h Kr^2 cos 2ö + Lr cos ø + Ì = 0,
rm sin ôçö + Arm~x sin(m - \)ö + ••• + Kr^2 sin 2ö + Lr sin ø = 0.What remained was to show that there actually were points where the two
curves intersected. For that purpose, Gauss divided both equations by rm and
argued that for large values of r the two curves must have zeros near the zeros of
cos my = 0 and sinmy? = 0. That would mean that on a sufficiently large circle,
each would have 2m zeros, and moreover the zeros of one curve, being near the
points with polar angles (fc + \/2)ð/ôç must separate those of the other, which are
near the points with polar angles kw/m. Then, arguing that the portion of each
curve inside the disk of radius r was connected, he said that it was obvious that
one could not join all the pairs from one set and all the pairs from the other set
using two curves that do not intersect.
Gauss was uneasy about the intuitive aspect of the proof. During his lifetime
he gave several other proofs of the theorem that he regarded as more rigorous.(^4) This restriction involves no loss of generality (see Problem 15.1).