The Fundamental Theorem Of Algebra Essay Research

The Fundamental Theorem Of Algebra Essay, Research Paper

The Fundamental Theorem of Algebra states that every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. In fact, there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors.

Early studies of equations by al’Khwarizmi (c 800) only allowed positive real roots and the Fundamental Theorem of Algebra was not relevant. Cardan was the first to realize that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x = 15x + 4 gave an answer getting -121, yet, Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his “complex numbers” to obtain the right answer yet he in no way understood his own math.

Bombelli, in his Algebra , published in 1572, was to produce a proper set of rules for manipulating these “complex numbers”. Descartes in 1637 says that one can “imagine” for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.

Viete gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L’invention en algebre . However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C. In fact this was to become the whole problem of the Fundamental Theorem of Algebra for many years since mathematicians accepted Albert Girard’s assertion as self-evident. They believed that a polynomial equation of degree n must have n roots, the problem was, they believed, to show that these roots were of the form

a + bi, a, b real.

Now, Harriot knew that a polynomial which vanishes at t has a root x – t but this did not become well known until stated by Descartes in 1637 in La geometrie , so Albert Girard did not have much of the background to understand the problem properly.

A “proof” that the Fundamental Theorem of Algebra was false was given by Leibniz in 1702 when he stated that x + t could never be written as a product of two real quadratic factors. His mistake came in not realizing that i could be written in the form a + bi, a, b real.

Euler, in a 1742 correspondence with Nicolaus(II) Bernoulli and Goldbach, showed that the Leibniz counterexample was false.

D’Alembert in 1746 made the first serious attempt at a proof of the Fundamental Theorem of Algebra. For a polynomial f he takes a real b, c so that f(b) = c. Now he shows that there are complex numbers z and w so that

|z| < |c|, |w| < |c|.

He then states the process to converge on a zero of f. His proof has several weaknesses. First, he uses a lemma without proof which was proved in 1851 by Puiseau, but whose proof uses the Fundamental Theorem of Algebra. Secondly, he did not have the necessary knowledge to use a compact argument to give the final emphasis. Despite this, the ideas in this proof are important. Euler was soon able to prove that every real polynomial of degree n, n 6 had exactly n complex roots. In 1749 he attempted a proof of the general case, so he tried to proof the Fundamental Theorem of Algebra for Real Polynomials:

Every polynomial of the nth degree with real coefficients has precisely n zeros in C.

His proof in Recherches sur les racines imaginaires des equations is based on decomposing a monic polynomial of degree 2 into the product of two monic polynomials of degree m = 2. Then since an arbitrary polynomial can be converted to a monic polynomial by multiplying by ax. for some k the theorem would follow by iterating the decomposition. Now Euler knew a fact which went back to Cardan in Ars 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

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