This post assumes familiarity with some basic concepts in
abstract algebra, specifically the terminology of field
extensions, and the classical results in Galois theory and group
theory. The fundamental theorem of algebra has quite a few
number of proofs (enough to fill a book!). In fact, it seems a
new tool in mathematics can prove its worth by being able to
prove the fundamental theorem in a different way. This series of
proofs of the fundamental theorem also highlights how in
mathematics there are many many ways to prove a single theorem,
and in re-proving an established theorem we introduce new
concepts and strategies. And perhaps most of all, we accentuate
the unifying beauty of mathematics. Problem: Let be a non-
constant polynomial. Prove has a root in . Solution: Without
loss of generality, we may assume has real coefficients, since
if has no roots, then neither does , where the bar represents
complex conjugation. In particular, is invariant under complex
conjugation, and all such quantities are real.