A rootless polynomial
The fundamental theorem of algebra proved by Carl Friedrich Gauss in
the last year of the 18th century states the nonexistence of some objects,
as many other important theorems of mathematics do. Namely, it states that
there is no nonconstant polynomial of the complex variable which vanishes
at no point of the complex plane C. (Those who have never heard
of complex numbers may wish to read Z. Marciniak's article about theedimensional
multiplication first.)
Up to this day many different proofs of Gauss's fundamental theorem
have been conceived, each of them of great beauty. Unfortunately, none
of them can be presented here with all the details. Nevertheless,
to satisfy the curious, here are the main ideas of two different proofs
which use no algebraic methods at all.
The first proof uses two lemmas
which immediately imply the fundamental theorem of algebra.
Lemma 1. If
P:C > C is a polynomial, then there is a point z_{0}
in C at which the function P attains its infimum.
The proof stems from the observation that for points z lying beyond
the closed circle K(0,R) for sufficiently large R the modulus of the polynomial
is quite big, so the infima of the two sets
are equal. The function P being continuous, it attains its infimum
on the compact set K(0,R).
Lemma 2. Let P be a polynomial of positive
degree. If w is such a complex number that P(w) is smaller than or equal
to P(z) for all complex z, then P(w)=0.
The proof of this lemma is more difficult and technical, but only elementary
knowledge about complex numbers is required.
The easiest way to proceed is to apply reductio ad absurdum.
Multipliying the polynomial by a constant and moving its graph on the plane
we can assume that w=0 and P(w)=a_{0}>0. Assume moreover that
the polynomial P contains no positive powers of z with exponent less than
k. Hence
Writing
for
(with z sufficiently small) we get
Now it is not hard to prove that by including the remaining parts of
the polynomial in the absolute value nothing is spoiled: there are finitely
many of them and in the neighborhood of zero their significance is infinitely
less than that of a_{k} z^{k}. We leave to the persistent
reader the determination of the exact value of "delta".
The second proof uses the socalled
Liouville's theorem. Let f:C >
C be a bounded function. If f is differentiable at every point,
i.e. for each z_{0 }in C there is a finite limit
then f is identically equal to a constant.
The proof of this fact can be found in any book on complex function
theory. It may be useful to observe, though, that Liouville's theorem becomes
false in the world of real numbers (consider e.g. f(x)=1/(x^{2}+1).
Let's assume now that P is a complex polynomial such that P(z) is nonzero
for all z. Then it can be observed that the function given by f(z)=1/P(z)
satisfies all the assumptions of Liouville.s theorem (to see that it is
bounded note, as in the proof of Lemma 1, that the modulus of the polynomial
cannot be small beyond some large circle). Hence 1/P is constant and consequently
also P is constant, which completes the proof of the fundamental theorem
of algebra.
P.S.
