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Somehow after two years in school. I feel like I have returned to square one. Still the directionless mathematician. Been reading alot recently, completed a proof on the fundamental theorem of algebra.

Gosh its been so long since I did any Maths here.

The fundamental theorem of algebra states that any polynomial in complex coeeficients has at least one complex root.

Its a simple collorary then that a polynomial of degree n will have n complex roots, since we can divide the original polynomial by the factor x-r, where r is the root, to get another polynomial of degree n-1, which will itself have at least one root.

The proof sketch hinges on the fact that a general polynomial, hereafter refered to as f(z), where f(z) = az^n + bz^(n-1) + ... + k , where a,b,...,k are complex;

1) is smooth, which means that as z changes gradually, so does f(z).
2) can be approximated by az^n for large z (meaning that the modulus of z is large)
3) is close to the constant term k for small z (meaning that the modulus of z is small)

First we consider the image of f(z) in the complex plane when modulus of z is large. It will trace a large circle about the origin of radius az^n as all the other terms in the polynomial become relatively small.

Then we consider the image when the modulus of z is small, it will trace a small circle about the constant term k, as all the other terms in the polynomial become small. We can also, assume without loss of generality that k is not zero.

Hence as the value of z shifts throughout the complex plane, decreasing in magnitude, the image of z will deform from the large circle about the origin, to the small cicle about the constant term k smoothly. In doing so, it must cross the origin at least once. Hence there must be a value of z which the image is zero, meaning that f(z) must be zero for at least one value of z, which completes the proof sketch.

I think this one is due to Gauss, one of the many proofs of this that he supplied during his lifetime.

It is regretful to think that even Euler couldn't figure this one out.