These questions can all be answered by the relevant literature.
1 (# of proofs by Gauss):
Actually, I have found the following statement: "Fuer diesen
beruehmten, tiefliegenden Existenzsatz gibt es viele Beweise; allein Gauss
lieferte acht. Alle Beweise benutzen indess nichtalgebraische Hilfsmittel."
The second statement refers to the topological question and is indeed true
as will be seen later, but the number of proofs by eight given there is
incorrect.
Next, I have gone through the book by B. L. van der Waerden:
Algebra, vierte Auflage DER MODERNEN ALGEBRA, 1955, Springer, Baende I, II.
Volume I, § 70 (Der Koerper der komplexen Zahlen) contains the following
statement: "Gauss hat fuer den Fundamentalsatz mehrere Beweise gegeben.
Den zweiten Gaussschen Beweis, der nur die einfachsten Eigenschaften der
reellen und komplexen Zahlen benutzt, dafuer aber recht schwere
algebraische Hilfsmittel heranzieht, werden wir in § 71 kennenlernen." In a
footnote van der Waerden refers to a proof of FTA by Jordan and one given
by Weyl on the basis of intuitionism. That is all van der Waerden said in
my (German) editon concerning the number of proofs of FTA by Gauss.
That second proof of FTA is indeed significant and was given in
1815 in the paper "Demonstratio nova altera theorematis omnem functionem
algebraicum rationalem integram unius variabilis in factores reales primi
vel secundi gradus resolvi posse". In this proof Gauss applied the
algebraic properties of symmetric functions, a differential equation between
the poynomial under consideration and its discriminant together with the
means van der Waerden mentioned in the above quote. This kind of argument
leads to the modern abstract proof of the existence of splitting fields
and the fact that this field is contained in the field of complex numbers -
and thats the reason why van der Waerden emphasized the second proof of
Gauss, only, and indicated § 71.
Until now I have dealt with wrong statements concerning the number
of proofs Gauss is supposed to be given.
Until now we know the following facts concerning the correct number
of proofs by Gauss.
Proof number 1: 1799 dissertation "Demonstratio nova theorematis omnem
functionem algebraicum rationalem integram unius variabilis in factores
reales primi vel secundi gradus resolvi posse"
Proof number 2: 1815 in the paper "Demonstratio nova altera theorematis
omnem functionem algebraicum rationalem integram unius variabilis in
factores reales primi vel secundi gradus resolvi posse"
Remark: The two titels only differ by altera.
Proof number 3: 1816 in the paper "Theorematis de resolubilitate
functionum algebraicarum integrarum in factores reales demonstratio tertia"
Proof number 4: 1849 on the occassion of his goldenes Doctorjubilaeum Gauss
came back to proof number 1. See also A. Ostrowski: Ueber den ersten und
vierten Beweis des Fundamentalsatzes der Algebra; in collected works of
Gauss.
Gauss died six years later in 1855.
The best thing to do is to checking through the respective parts of the
collected works of Gauss and of Oswald's Klassiker der exakten
Wissenschaften, namely Nr. 14: Die vier Gauss'schen Beweise fuer die
Zerlegung ganzer algebraischer Funktionen ersten und zweiten Grades,
Akademische Verlagsanstalt m. b. H., Leipzig und Berlin 1913.
Nevertheless there are other serious sources which endorse that Gauss gave
four proofs; I am going to give some of them:
[Klei 1956] Felix Klein: Vorlesungen ueber die Entwicklung der Mathematik im
19. Jahrhundert, New York, 1956, pp.25 - 28. There he gives references to
the collected works. Klein was born in 1849 when Gauss was still alive.
[Bueh 1981] Walter Buehler: Gauss. Eine biographische Studie. Berlin, 1981.
On pp. 40 - 42 he comments on each of the four proofs. In a letter
(16.12.1799) Gauss resumed the contents of his dissertation, p. 40/41. In
here, he was critical of his predicessors. There is an English edition.
References are given.
[Nar 1978] W. Narkiewics: The work of C. F. Gauss in algebra and number
theory, 75 - 82. In Abhandlungen der Akademie der Wissenschaften der DDR,
Abteilung Mathematik, Naturwissenschaften, Technik. Festakt und Tagung aus
Anlass des 200. Geburtstages von Carl Friedrich Gauss; 22./23. April 1977
in Berlin. Berlin, 1978. W. Nakiewics stated proof number 4 was dated 1850.
I guess this was the date of publication while 1849 was the date of the
special day (goldene Doktorpromotion), on occassion of which Gauss lectured
on FTA.
[Dieu 1985] Jean Dieudonne': Geschichte der Mathematik 1700 - 1900. Ein
Abriss. Wiesbaden, 1985. (There exists a French edition: Abre\ge' d'histoire
des mathematiques 1700 - 1900, Tome I, II), p. 57, 69ff, 135f.
2 (the question of rigor):
This question is dealt with in [Dieu 1985].
Relating to proof number 1, it shows that in lack of clear theorems in
analysis (no implicit function theorem, ...) the continuation of branches
of curves became critical. Dieudonne' concludes, p. 70/71: "kann man nicht
sagen, dieser Beweis genuege voellig den heutigen Anforderung an Strenge;
es ist aber durchaus moeglich, die Gausssche Ueberlegung so zu gestalten,
daß sie voellig korrekt und streng ist." He indicates A. Ostrowski; see
quotation above. Relating to proof number 2 and 3 Diendonne' says: "Der
zweite und der dritte Beweis von Gauss dagegen sind voellig streng."
Remember that proof number 2 was emphasized by B. L. van der Waerden in his
algebra book.
3 (the question of topology or continuity in the proofs of FTA):
Mr. Martin Davis and Mr. Eric Detrez are in general not wrong when they
state, that there is left an irreducible piece of analysis. I am not sure
whether my arguments are of any use to Mr. Gordon Fisher's considerations.
All proofs of the theorem involve some form of topological properties of
real or complex numbers.The role of topology has ultimately been reduced to
the single assumption that a polynomial of odd degree with real
coefficients has a real root. This is dealt with in the theory of real
fields (Algebra book of Serge Lang (second edition, 1984), ch. XI (Real
Fields due to E. Artin and O. Schreier)) or the above quoted algebra book
by van der Waerden, § 71, Satz 3a: Besitzt in einem angeordneten Koerper K
jedes positive Element eine Quadratwurzel und jedes Polynom ungraden Grades
mindestens eine Nullstelle, so ist der durch Adjunktion von i entstehende
Koerper algebraisch abgeschlossen. In case K is the field of complex
numbers, theorem 3a is the FTA. In Lang consider theorem 2.2 and remember
that a real closed field is not algebraic closed, but produces this
property by adjunction of i (Satz 3a). This way FTA results from theorem
2.2. What is significant with this, is the fact, that by the theory of real
fields or as it was called in the late twenties "der Theorie der
<angeordneten> Koerper" created by E. Artin and O. Schreier in 1927
(observe the terminologie of van der Waerden's book based on lectures by
Artin and Noether) the role of continuity or what is the same of topology
in the proof of the FTA was completely analysed, [Dieu 1985], p. 70. It is
now plausible that "No elementary algebraic proof of this theorem exists,
The new Enzyklopaedia Britannica in 30 volumes, Chicago, ..., 1984, 1:504"
The determination and estimation of zeros of polynomials as well as the
proofs of the FTA make more use of the theory of functions than of algebra.
Therefore, Pa/l Tura/n (C. r. I. Congr. Math. Hongr. 1950, 279 - 290 (1952))
suggested to denote this part of algebra Funktionalalgebra. See in this
context the papers by Bashmakova, I. G.: On a proof of the fundamental
theorem of algebra, Istor. Mat. Issled., no. 10 (1957), 257 - 304 (in
Russian).
Arnold, B. H.: A topological proof of the fundamental theorem of algebra.
Amer. Math. Monthly 56 (1949), 465 - 466. This proof is based on the
Brouwer fixed-point theorem.
Gerhard Warnecke