Editing Fundamental theorem of algebra (section)

==History==
{{ill|Peter Roth (mathematician)|lt=Peter Roth|de|Peter Roth (Mathematiker)}}, in his book ''Arithmetica Philosophica'' (published in 1608, at Nürnberg, by Johann Lantzenberger),<ref>[http://www.e-rara.ch/doi/10.3931/e-rara-4843 Rare books]</ref> wrote that a polynomial equation of degree ''n'' (with real coefficients) ''may'' have ''n'' solutions. [[Albert Girard]], in his book ''L'invention nouvelle en l'Algèbre'' (published in 1629), asserted that a polynomial equation of degree ''n'' has ''n'' solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds "unless the equation is incomplete", where "incomplete" means that at least one coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation <math>x^4 = 4x-3,</math> although incomplete, has four solutions (counting multiplicities): 1 (twice), <math>-1+i\sqrt{2},</math> and <math>-1-i\sqrt{2}.</math>

As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degrees are either 1 or 2. However, in 1702 [[Gottfried Leibniz|Leibniz]] erroneously said that no polynomial of the type {{math|''x''<sup>4</sup> + ''a''<sup>4</sup>}} (with {{math|''a''}} real and distinct from 0) can be written in such a way. Later, [[Nicolaus I Bernoulli|Nikolaus Bernoulli]] made the same assertion concerning the polynomial {{math|''x''<sup>4</sup> − 4''x''<sup>3</sup> + 2''x''<sup>2</sup> + 4''x'' + 4}}, but he got a letter from [[Leonhard Euler|Euler]] in 1742<ref>See section ''Le rôle d'Euler'' in C. Gilain's article ''Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral''.</ref> in which it was shown that this polynomial is equal to
:<math>\left (x^2-(2+\alpha)x+1+\sqrt{7}+\alpha \right ) \left (x^2-(2-\alpha)x+1+\sqrt{7}-\alpha \right ),</math>
with <math>\alpha = \sqrt{4+2\sqrt{7}}.</math>
Euler also pointed out that
:<math>x^4+a^4= \left (x^2+a\sqrt{2}\cdot x+a^2 \right ) \left (x^2-a\sqrt{2}\cdot x+a^2 \right ).</math>

A first attempt at proving the theorem was made by [[Jean le Rond d'Alembert|d'Alembert]] in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as [[Puiseux's theorem]]), which would not be proved until more than a century later and using the fundamental theorem of algebra. Other attempts were made by [[Leonhard Euler|Euler]] (1749), [[François Daviet de Foncenex|de Foncenex]] (1759), [[Joseph Louis Lagrange|Lagrange]] (1772), and [[Pierre-Simon Laplace|Laplace]] (1795). These last four attempts assumed implicitly Girard's assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was ''a''&nbsp;+&nbsp;''bi'' for some real numbers ''a'' and ''b''. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a [[splitting field]] of the polynomial ''p''(''z'').

At the end of the 18th century, two new proofs were published which did not assume the existence of roots, but neither of which was complete. One of them, due to [[James Wood (mathematician)|James Wood]] and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof had an algebraic gap.<ref>Concerning Wood's proof, see the article ''A forgotten paper on the fundamental theorem of algebra'', by Frank Smithies.</ref> The other one was published by [[Carl Friedrich Gauss|Gauss]] in 1799 and it was mainly geometric, but it had a topological gap, only filled by [[Alexander Ostrowski]] in 1920, as discussed in Smale (1981).<ref>[https://www.semanticscholar.org/paper/The-fundamental-theorem-of-algebra-and-complexity-Smale/bc3d674b9931e49d1e023d16401ae15ee6ad6681 Smale writes], "...I wish to point out what an immense gap Gauss's proof contained. It is a subtle point even today that a real algebraic plane curve cannot enter a disk without leaving. In fact, even though Gauss redid this proof 50 years later, the gap remained. It was not until 1920 that Gauss's proof was completed. In the reference Gauss, A. Ostrowski has a paper which does this and gives an excellent discussion of the problem as well..."</ref>

The first rigorous proof was published by [[Jean-Robert Argand|Argand]], an [[List of amateur mathematicians|amateur mathematician]], in 1806 (and revisited in 1813);<ref>{{MacTutor Biography|id=Argand|title=Jean-Robert Argand}}</ref> it was also here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 1816 and another incomplete version of his original proof in 1849.

The first textbook containing a proof of the theorem was [[Cauchy]]'s ''[[Cours d'Analyse|Cours d'analyse de l'École Royale Polytechnique]]'' (1821). It contained Argand's proof, although [[Jean Robert Argand|Argand]] is not credited for it.

None of the proofs mentioned so far is [[Constructivism (mathematics)|constructive]]. It was [[Weierstrass]] who raised for the first time, in the middle of the 19th century, the problem of finding a [[constructive proof]] of the fundamental theorem of algebra. He presented his solution, which amounts in modern terms to a combination of the [[Durand–Kerner method]] with the [[homotopy continuation]] principle, in 1891. Another proof of this kind was obtained by [[Hellmuth Kneser]] in 1940 and simplified by his son [[Martin Kneser]] in 1981.

Without using [[countable choice]], it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the [[construction of the real numbers|Dedekind real numbers]] (which are not constructively equivalent to the Cauchy real numbers without countable choice).<ref>For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; <cite>A weak countable choice principle</cite>; available from [http://math.fau.edu/richman/HTML/DOCS.HTM] {{Webarchive|url=https://web.archive.org/web/20200219002009/http://math.fau.edu/richman/html/docs.htm|date=2020-02-19}}.</ref> However, [[Fred Richman]] proved a reformulated version of the theorem that does work.<ref>See Fred Richman; 1998; <cite>The fundamental theorem of algebra: a constructive development without choice</cite>; available from [http://math.fau.edu/richman/HTML/DOCS.HTM] {{Webarchive|url=https://web.archive.org/web/20200219002009/http://math.fau.edu/richman/html/docs.htm|date=2020-02-19}}.</ref>