The Fundamental Theorem Of Algebra Essay Research — страница 2

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Magna, or earlier, that a transformation could be applied to remove the second largest degree term of a polynomial. Therefore, he assumed that x + Ax + Bx +… =(x + tx + gx + …)(x – tx + hx + …) and then multiplied and compared coefficients. Euler claimed to g led to h, … being rational functions of A, B, …, t. All this was carried out in detail for n = 4, but the general case is only a sketch. In 1772 Lagrange raised objections to Euler’s proof. He stated that Euler’s rational functions could lead to 0/0. Lagrange used his knowledge of permutations of roots to fill all the gaps in Euler’s proof except that he was still assuming that the polynomial equation of degree n must have n roots of some kind so he could work with them and deduce properties, like

eventually that they had the form a + bi, a, b real. Laplace, in 1795, tried to prove the Fundamental Theorem of Algebra using a completely different approach using the discriminant of a polynomial. His proof was very elegant and its only ‘problem’ was that again the existence of roots was assumed. Gauss is usually credited with the first proof of the Fundamental Theorem of Algebra. In his doctoral thesis of 1799 he presented his first proof and also his objections to the other proofs. He is undoubtedly the first to spot the fundamental flaw in the earlier proofs, namely the fact that they were assuming the existence of roots and then trying to deduce properties of them. Of Euler’s proof Gauss says “… if one carries out operations with these impossible roots, as though

they really existed, and says for example, the sum of all roots of the equation x+ax+bx+.” = 0 is equal to -a even though some of them may be impossible (which really means: even if some are non-existent and therefore missing), then I can only say that I thoroughly disapprove of this type of argument. Gauss himself does not claim to give the first proper proof. He merely calls his proof new but says, for example of d’Alembert’s proof, that despite his objections a rigorous proof could be constructed on the same basis. Gauss’s proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our present day standards required for a rigorous proof. In 1814 the Swiss accountant Jean Robert Argand published a proof of the Fundamental Theorem of Algebra

which may be the simplest of all the proofs. His proof is based on d’Alembert’s 1746 idea. Argand had already sketched the idea in a paper published two years earlier Essai sur une maniere de representer les quantities imaginaires dans les constructions geometriques . In this paper he interpreted i as a rotation of the plane through 90 so giving rise to the Argand plane or Argand diagram as a geometrical representation of complex numbers. Now in the later paper Reflexions sur la nouvelle theorie d’analyse Argand simplifies d’Alembert’s idea using a general theorem on the existence of a minimum of a continuous function. In 1820 Cauchy was to devote a whole chapter of Cours d’analyse to Argand’s proof. This proof only fails to be rigorous because the general concept

of a lower bound had not been developed at that time. The Argand proof was to attain fame when it was given by Chrystal in his Algebra textbook in 1886. Chrystal’s book was very influential. Two years after Argand’s proof appeared Gauss published in 1816 a second proof of the Fundamental Theorem of Algebra. Gauss uses Euler’s approach but instead of operating with roots which may not exist, Gauss operates with indeterminates. This proof is complete and correct. A third proof by Gauss also in 1816 is, like the first, topological in nature. Gauss introduced in 1831 the term ‘complex number’. The term ‘conjugate’ had been introduced by Cauchy in 1821. Gauss’s criticisms of the Lagrange-Laplace proofs did not seem to find immediate favor in France. Lagrange’s 1808

2nd Edition of his treatise on equations makes no mention of Gauss’s new proof or criticisms. Even the 1828 Edition, edited by Poinsot, still expresses complete satisfaction with the Lagrange-Laplace proofs and no mention of the Gauss criticisms. In 1849 (on the 50th anniversary of his first proof) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots. The proof is similar to the first proof given by Gauss. However it is adds little since it is straightforward to deduce the result for complex coefficients from the result about polynomials with real coefficients. It was in searching for such generalizations of the complex numbers that Hamilton discovered the quaternions around 1843, but of course the quaternions are