How can the fundamental theorem of algebra be related to ano

How can the fundamental theorem of algebra be related to another area in math which would be suitable for an undergraduate dissertation

Solution

Another field in which fundamental theorem of Algebra can be used is polynomials. It helps decide the nature of roots of polynomial.

Gauss’s doctoral dissertation provides the first genuine proof of the fact that every polynomial in one variable with real coefficients can be factored into linear or quadratic factors. Imaginary and complex numbers were not widely accepted at that time, but presently this proposition which was initially called the fundamental theorem of algebra is presently expressed by saying that every polynomial of degree n possesses n complex roots, counting multiplicities. Although Gauss’s proof focused on polynomials with real coefficients, it isn’t difficult to extend the result to polynomials with complex coefficients as well. By modern standards, Gauss’s proof was not rigorously complete, since he relied on the continuity of certain curves, but it wasa major improvement over all previous attempts at a proof.Only about a third of Gauss’s dissertation was actually taken up by the proof. The rest consisted of a rather frank assessment of the previously claimed proofs of this proposition by D’Alembert, Euler, Legendre, Lagrange, and others. Gauss explained that all their attempts were fallacious, and indeed that they didn’t even address the real problem. They all implicitly assumed the existence of the roots, and just sought to determine the formof those roots. Gauss pointed out that the real task was to prove theexistenceof the roots in the first place, and only then to establish their form. It’s clear that Gauss attached great importance to this theorem, since he returned to it repeatedly, publishing four proofs of it during his lifetime. The fourth of these was in the last paper he ever wrote, which appeared in 1849, exactly 50 years after his dissertation on the same subject.The phrase “fundamental theorem of algebra” may be considered somewhat inaccurate, because the proposition is actually part of complex analysis. (It might even be regarded as an example of Gödel’s incompleteness theorem, i.e., a meaningful and true proposition that cannot be proven within the contextin which it is formulated.) A huge number of proofs have been devised, making use of a wide range of mathematical concepts.