I am grateful to Sanford Segal for his kind comments on my contribution.
The non-algebraic portion of the FTA over the complex numbers is based on
the Weierstrassschen Nullstellensatz fuer Polynome which is the
Weierstrasssche Nullstellensatz for continuous functions restricted to
polynomials. The polynomials of odd degree satisfy the boundary conditions
in the Weierstrassschen Nullstellensatz for continuous functions
respectively for polynommials and this result is therefore looked at the
topological part of FTA. B. L. van der Waerden writes in his algebra book:
"Der Weierstrasssche Nullstellensatz fuer Polynome ist das Fundament aller
Saetze ueber die reellen Wurzeln algebraischer Gleichungen. Wir werden ihn
spaeter auf andere Koerper als den der reellen Zahlen, naemlich auf die
sog. <reell abgeschlossenen Koerper> ausdehnen. Alle weiteren Saetze dieses
Paragraphen beruhen ausschliesslich auf dem Weierstrassschen
Nullstellensatz fuer Polynome und gelten dementsprechend auch für die
spaeteren allgemeineren Koerper (vol I, § 69 of B. L. van der Waerden:
Algebra, vierte Auflage DER MODERNEN ALGEBRA, 1955, Springer, 2 Baende)."
Actually, only continuity is needed for proof; nevertheless the
Weierstrasssche Nullstellensatz is also a fundamental theorem of analysis,
but it is the calculus which figures the quintessence of analysis und not
continuity while continuity figures the quintessence of topolgy. See the
access to topology via the concept of limit points in [HocYou 1961] (John
G. Hocking, Gail S. Young: Topology, Reading, ... 1961). These authors
write in the introduction: "The very definition of a continuous function is
an example of this dependence (on the properties of limit points). To
exaggerate, one might view topology as the complement of modern algebra in
that together they cover the two fundamental types of operations found in
mathematics." This corresponds to what van der Waerden said in the above
quotation. To emphasize his point of view a bit more, again van der Waerden
will help (vol I, chapter 9, reelle Koerper): "Beim Studium der
algebraischen Zahlkoerper spielen ausser den algebraischen Eigenschaften
ihrer Zahlen gewisse unalgebraische Eigenschaften: absolute Betraege |*|,
Realitaet, Positivsein, eine Rolle. Dass diese Eigenschaften sich nicht mit
Hilfe der algebraischen Operationen + und * eindeutig definieren lassen,
zeigt sich an folgendem Beispiel. ..." So there are arguments contra et
pro. Ta/l Tura/n's proposal to call Funktionalalgebra that part of algebra
dealing with FTA (note that Gauss' first proof needs the implicit function
theorem for proof) etc. because of the significant use of functions there,
this finally shows that in the matter of names Professor Faust may be right
as Sandford Segel emphasized: Name ist Schall und Rauch (Names mean
nothing) (Goethe, Faust, part I, Gretchenszene) as to whether the
unavoidable "non-algebraic" portion of the usual FTA over the complex
numbers is called analysis or topology seems irrelevant.
Gerhard Warnecke
Remark: There is an English edition of the van der Waerden algebra book. So
anybody willing to read the above quotations in English can do. I think the
play Faust by Goethe is translated into nearly every civilized language.
Gerhard Warnecke