The fundamental theore of algebra

The fundamental theorem of Algebra states that every polynomial of degree larger than zero $$ p(x) = a_0 + a_1 x + … + a_nx^n \qquad (n>0) $$ has at least one complex root. Obviously the “theorem” is not true with real numbers (the polynomial \( x^2+1\) has no real roots). The first ste in this project is to get used to the complex numbers. Real numbers are represented by points in a line, and in a similar way complex numbers are represented by points in a plane. As such, they can be represented by the Cartesian coordinates, or by polar coordinates (you will need to understand this last sentence).

The fundamental theorem of algebra is of tremendous importance for “higher maths”, but it does not have many attractive applications to make it shine (at least that I am aware of). So this project should focus in understanding a/many proof(s) of the theorem. The minimum is to understand one of the proofs.

It is not easy to find a source where a proof of the fundamental theorem of algebra is explained in clear terms. Probably the only option is to go for the proof of Birkhoff and MacLane. This proof relies on the fact that any polynomial $$ p(x) = a_0 + a_1 x + … + a_nx^n \qquad (n>0) $$ is, when \( |x|\ll 1 \), very similar to $$ a_0 + a_1 x,, $$ and this is a circle that winds once around the point \(a_0\), arbitrarily close to it.

On the other hand, for \( |x|\gg 1 \), the polynomial \( p(x) \) is very similar to $$ a_nx^n,, $$ and this is a circle that winds \( n\) times around the origin.

Once you understand these two facts, it is easy to see that by changing \( x\) from a very small to a very large number (in a continous way), you must cross the origin (and there is your root).

The wikipedia has a lot of information, but probably the most concise and useful information for this project is in cut-the-knot.org.