[Math] Fundamental theorem of algebra: a proof for undergrads

Question

The fundamental theorem of algebra is the statement that a complex polynomial of positive degree has at least one root. I do not know complex analysis but I searched for proofs of the statement and came across proofs using complex analysis which seemed rather short and elegant. This is to say: I am aware that there exist very easy proofs using tools of complex analysis. Since I do not know complex analysis yet I started to wonder if there are any known proofs that are short and easy that use only tools that a first year undergrad knows? Like real analysis and linear algebra?

Answer 1

Here are three accessible proofs, via Keith Conrad:

http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/fundthmalgcalculus.pdf

http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/fundthmalglinear.pdf

http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/propermaps.pdf

(the last one requires a bit more sophistication, but isn't too bad)