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One of the most intriguing aspects of mathematics is the connectivity between its branches. A great example is the topological proof of the Fundamental Theorem of Algebra.

The Fundamental Theorem of Algebra states that any nonconstant polynomial has a root in the complex plane. In fact, a degree n polynomial has n roots, if you count roots with multiplicity, but it suffices to show that there is at least one root. The Fundamental Theorem of Algebra was proved by Gauss in his doctoral dissertation.

To prove this using topology, we need the idea of the winding number of an oriented curve in the plane about the origin. This is a precursor to the fundamental group of a space. The winding number of a curve which does not pass through the origin is an integer which counts how often the curve goes around the origin in the clockwise direction. The winding number does not change under continuous deformations through curves which do not pass through the origin.

If the constant term of the polynomial is 0, then 0 is a root of the polynomial, and we do not need to prove anything more. So, let use assume the constant term of the polynomial is c != 0.

Consider the image of circles about the origin under the polynomial. Small circles will be mapped into a small neighborhood of c, and will thus not wind around the origin at all. The images of small circles have winding number 0. Sufficiently large circles will behave like the largest term, and can be deformed to their image under z|->zn, where n is the degree of the polynomial. That winds n times around the origin.

As we increase the radius from 0 to infinity, the image changes continuously, and the winding number changes from 0 to n. Since the winding number is preserved under deformations which do not pass through the origin, and n>0, there must be some radius r so that the image of the circle of radius r does pass through the origin, so there must be some z with |z|=r so that the polynomial sends z to 0, a root. QED

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