viernes, 14 de octubre de 2016

Yet another function-theoretic proof of the Fundamental Theorem of Algebra

Let $f$ be a nonconstant polynomial with complex coefficients. Since $|f(z)| \to \infty$ as $z \to \infty$, we guarantee the existence of $R>0$ such that $$|f(z)|>|f(0)| \quad \quad (\ast)$$ for every $z \in \mathbb{C} \setminus \mathrm{B}_{R}(0)$. On the other hand, the continuity of the function $F \colon \overline{\mathrm{B}_{R}(0)} \to \mathbb{C}$ given by $z \overset{F}{\longmapsto} |f(z)|$ and the compactness of $\overline{\mathrm{B}_{R}(0)}$ allow us to ascertain the existence of $z_{0} \in \overline{\mathrm{B}_{R}(0)}$ such that $$|f(z_{0})| \leq |f(z)|$$ for every $z \in \overline{\mathrm{B}_{R}(0)}$. From $(\ast)$ we infer that $z_{0}$ is actually an element of $\mathrm{B}_{R}(0)$; then, by resorting to the Minimum-Modulus Principle, we conclude that $|f(z_{0})|$ must be equal to $0$ and we are done.

Scholia. a) If I understand correctly, the basic idea in this approach to the Fundamental Theorem of Algebra can be traced back to a 1748 memoir of d' Alembert. Yet, according to what we read in Reinhold Remmert's essay on the Fundamental Theorem of Algebra in [1, pp. 99-122], there were some gaps in d' Alembert's original argument that would be pointed out by a twenty-two-year-old Gauss in the beginning of his doctoral thesis "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse" which he submitted to Pfaff at the University of Helmstedt in 1799 and through which he obtained his doctorate. However, it is noteworthy that, on that occasion, "... Gauss also [remarked], almost prophetically (Werke 3, p.11): 'For these reasons I am unable to regard the proof by d' Alembert as entirely satisfactory, but that does not prevent, in my opinion, the essential idea of the proof from being unaffected, despite all objections; I believe that ... a rigorous proof could be constructed on the same basis.'"
b) Interestingly enough, the proof of the Fundamental Theorem of Algebra showcased by Aigner & Ziegler's in their Proofs from THE BOOK (5th. edition, pp. 147-149) is based on the aforementioned d'Alembertian attack as subsequently simplified by Argand in 1814.

[1] H. D. Ebbinghaus, et al., Numbers. Graduate Texts in Mathematics 123, Springer-Verlag, NY, 1991.