In a recent email Professor Diego Vaggione of the National University of Cordoba, Argentina kindly drew my attention to a note of his that appeared in Colloquium Mathematicum not long ago. The note that presents a short proof of the Fundamental Theorem of Algebra follows (in an HTML rendition) the message from Professor Vaggione.
Dear professor Bogomolny:
I have visited your web site on mathematics. I found it very interesting. I am enclosing a latex file of my paper "On the Fundamental Theorem of Algebra" (Colloquium Mathematicum, Vol. 73, No. 2 (1997), 193-194) in which I show that the clasical proof of the Fundamental Theorem of Algebra via Liouville can be substantialy simplified. Perhaps you can include this proof at your web site.
|VOL. 73||1997||NO. 2|
ON THE FUNDAMENTAL THEOREM OF ALGEBRA
DIEGO VAGGIONE (CÓRDOBA)
The difficulty presented by such a scheme is that the proofs of Liouville's theorem involve complex integration which makes the reader believe that a proof of the Fundamental Theorem of Algebra is too involved. even when topological arguments are used.
In this note we show that such a difficulty can be avoided by giving a simple proof of the Maximum Modulus Theorem for rational functions and then obtaining the Fundamental Theorem of Algebra as a corollary. The proof obtained in this way is intuitive and mnemotechnic in contrast to the usual elementary proofs of the Fundamental Theorem of Algebra.
As usual we use C to denote the set of complex numbers. By D(a, ε) we denote the set
LEMMA. Let f be a function such that f(D(a,ε)) is contained in a half plane whose defining straight line contains 0. Let
|1||= limn→ε f(z)/b(zn - a)k|
|= Re limn→ε f(z)/b(zn - a)k|
|= limn→ε Re f(z)/b(zn - a)k ≤ 0,|
which is absurd. Q.E.D.
MAXIMUM MODULUS THEOREM FOR RATIONAL FUNCTIONS. Let
Proof. Suppose that R is not constant. Since
(R(z) - R(a))/(z - a)k = p1(z)/[q(a)q(z)(z - a)k] = c(z)/[q(a)q(z)]
limz→a (R(z) - R(a))/(z - a)k ≠ 0.
Since |R(z)|≤|R(a)| for every z ∈ D(a, ε),
FUNDAMENTAL THEOREM OF ALGEBRA. A polynomial with no zeros is constant.
Proof. Suppose that p(z) is not constant and
Acknowledgement. I would like to thank María Elba Fasah for her assitance with the linguistic aspects of this note.
Facultad de Mathemática, Astronomía y Física (FAMAF)
Universidad Nacional de Córdoba
Córdoba 5000, Argentina
Received 8 August 1996