| On page 598 of
|
| MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN TIMES,
|
| Morris Kline writes
|
| The first substantial proof of the fundamental theorem [of algebra],
| though not rigorous by modern standards, was given by Gauss in his
| doctoral thesis of 1799 at Helmstadt. He criticized the work of
| d'Alembert, Euler, and Lagrange and then gave his own proof.
|
| On the same page, Kline writes
|
| Gauss gave three more proofs.
|
| However, on page 224 of Modern Algebra, English translation 1949, 1953,
| B. L. Van der Waerden writes
|
| Gauss gave five proofs of the fundamental theorem.
|
| My questions:
|
| Did Gauss give four proofs or five?
Dear Sam,
Gauss gave four proofs, which in fact appear on the first pages of
the third volume of his _Werke_. The first Gaussian proof [1] of the
Fundamental Theorem of Algebra [FTA] is considered also the *first*
satisfactory proof, and according to Gauss
"... hatte einen doppelten Zweck, nemlich erstens, zu zeigen,
dass sa"mmtliche bis dahin versuchte Beweise dieses wichtigsten
Lehrsatzes der Theorie der algebraischen Gleichungen ungenu"gend
und illusorisch sind, und zweitens, einen neuen vollkommen
strengen Beweis zu geben."
["... it had a double purpose; first of all, to show that previous proofs
of this most important theorem of the theory of algebraic equations
conceived so far are insufficient and illusory, and secondly, to give a
new perfectly rigorous proof."]
[1] _Demonstratio nova theorematis omnem functionem algebraicam rationalem
integram unius variabilis in factores reales primi vel secundi gradus
resolvi posse_ ("A New Proof of the Theorem that Every Integral Rational
Algebraic Function of One Variable can be Resolved into Real Factors of
the First or Second Degree"), published at Helmsta"dt, 1799; cf. _Werke_,
vol. 3 (1876), pp. 1-30.
As Kline put it, Gauss gave three more proofs:
[2] _Theorematis omnem functionem algebraicam rationalem integram unius
variabilis in factores reales primi vel secundi gradus resolvi posse_
(Dec. 7, 1815) published in _Commentationes societatis regiae scientiarum
Gottingensis recentiores_, vol. 3 (1816), pp. 107-134; cf. also _Werke_,
vol. 3 (1876), pp. 31-56.
[3] _Theorematis de resolubilitate functionum algebraicarum integrarum
in factores reales, demonstratio tertia, supplementum commentationis
praecedentis_ (Jan. 30, 1816) published in _Commentationes societatis
regiae scientiarum Gottingensis recentiores_, vol 3 (1816), pp. 135-142;
cf. also _Werke_, vol. 3 (1876), pp. 57-64.
[4] _Beitra"ge zur Theorie der algebraischen Gleichungen_ (July 16, 1849),
_Abhandlungen der Ko"nigl. Gessellschaft der Wissenschaften zu Go"ttingen_
vol. 4, pp. 3-15, Go"ttingen, 1850; cf. also _Werke_, vol. 3 (1876),
pp. 71-85.
| Are there any proofs by Gauss "rigorous by modern standards"?
|
| The easiest proof & the only one that I have ever studied, follows
| from Liouville's theorem, and if I remember correctly, it is steeped
| in analysis. Is it possible to prove the fundamental theorem of
| algebra strictly algebraically, with no element of analysis?
According to D.E. Smith [5], "while the first proof is based in part on
geometrical considerations, the second is entirely algebraic and has been
described as 'the most ingenious in conception and the most far-reaching
in method of the four' [6]. It is appropriate, therefore, to give here the
second proof". Because of the limitations of space, DES does present the
entire paper [2], passing over the introduction (parag. 1), and gives a
brief resume of paragraphs 2 through 6, which contain the proof of certain
theorems on the primality of rational integral functions, and on symmetric
functions, which are now well-known. From this point on, the translation
from the Latin into English is given in full except for one section. When
DES states that "the second is entirely algebraic" in fact he implies that
this Gaussian proof is not based on geometrical considerations; however,
analysis is used in the proof, as it is easy to see.
[5] Smith, D.E.: "A Source Book in Mathematics", New York: Dover
Publications, Inc., 2 vols., 1959. See vol. 1, pp. 292-306.
[6] Gauss, Carl F.; Netto, Eugen:
"Die Vier Gauss'schen Beweise fuer die Zerlegung ganzer algebraischer
Functionen in reele Factoren ersten oder zweiten Grades (1799-1849)"
["The four Gaussian proofs for the factoring of Integral Algebraic
Functions into Real Factors of First or Second Degree"] , by Carl F.
Gauss, Hrsg. von E. Netto, Leipzig: W. Engelmann, 1890. In the quote
mentioned above, DES refers to page 81 of the third edition of Netto's
book, published in Leipzig, in 1913.
By the way, Struik's "A Source Book in Mathematics" [7] contains the
frame of the first Gaussian proof (pp. 115-122). Struik remarks that
"in the geometrical language he [Gauss] uses he transfers geometrical
continuity to arithmetical quantities without proof, but at least at
one place expresses his conviction that he can make this aspect of his
proof also strictly rigorous. This can indeed be accomplished by the
methods of Bolzano anb Weierstrass".
[7] Struik, D.J.:
"A Source Book in Mathematics, 1200-1800", New Jersey: Princeton
University Press, 1969. See pp. 115-122.
There are tons of information on the history of FTA. For a bird's-eye
*online-view* of it, you may be interested in checking
http://www-groups.dcs.st-andrews.ac.uk/~history/HistoryTopics.html
| My planner from last January indicates that I have seen a book
| with the title
|
| The Fundamental Theorem of Algebra [FTA]
|
| by Fine and Rosenberger. Does anyone on the list know the merits
| of this text?
The purpose of Fine and Rosenberger textbook is to discuss a few proofs
of the FTA from different standpoints -- algebra, complex analysis, and
topology.
The following is the ToC of the book:
1. Complex numbers
2. Polynomials and complex polynomials
3. Complex analysis and analytic functions
4. Complex integration and Cauchy's theorem
5. Fields and field extensions
6. Galois theory
7. Topology and topological spaces
8. Algebraic topology and the final proof
Appendices
A. A version of Gauss's original proof
B. Cauchy's Theorem Revisited
C. Three additional complex analytic proofs of the FTA
D. Two More Topological Proofs of the FTA
With best wishes,
Julio