|
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree ''n'' polynomial with complex coefficients has, counted with multiplicity, exactly ''n'' roots. The equivalence of the two statements can be proven through the use of successive polynomial division. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients. ==History== Peter Rothe, in his book ''Arithmetica Philosophica'' (published in 1608), 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", by which he meant that no 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 ''x''4 = 4x − 3, although incomplete, has four solutions (counting multiplicities): 1 (twice), −1 + ''i''√2, and −1 − ''i''√2. 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 degree is either 1 or 2. However, in 1702 Leibniz said that no polynomial of the type ''x''4 + ''a''4 (with ''a'' real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial ''x''4 − 4''x''3 + 2''x''2 + 4''x'' + 4, but he got a letter from Euler in 1742〔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''.〕 in which he was told that his polynomial happened to be equal to : where α is the square root of 4 + 2√7. Also, Euler mentioned that : A first attempt at proving the theorem was made by 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 furthermore the proof assumed the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and 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'' + ''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. One of them, due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof had an algebraic gap.〔Concerning Wood's proof, see the article ''A forgotten paper on the fundamental theorem of algebra'', by Frank Smithies.〕 The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, filled by Alexander Ostrowski in 1920, as discussed in Smale 1981 () (Smale writes, "...I wish to point out what an immense gap Gauss' 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' 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..."). A rigorous proof was published by Argand in 1806; it was 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 version of his original proof in 1849. The first textbook containing a proof of the theorem was Cauchy's ''Cours d'analyse de l'École Royale Polytechnique'' (1821). It contained Argand's proof, although Argand is not credited for it. None of the proofs mentioned so far is 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, that 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 Dedekind real numbers (which are not constructively equivalent to the Cauchy real numbers without countable choice〔For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from ().〕). However, Fred Richman proved a reformulated version of the theorem that does work.〔See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from ().〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「fundamental theorem of algebra」の詳細全文を読む スポンサード リンク
|