the fundamental theorem of algebra

Every nonconstant polynomial with complex coefficients has at least one complex root. Equivalently, a degree-n polynomial has exactly n roots in the complex numbers, counted with multiplicity. This is the promise that the complex numbers are 'algebraically closed' — you never need to invent new numbers to solve a polynomial once you have allowed i. It is the reason the complex plane is the natural home of polynomial algebra.

Complex analysis gives a remarkably short proof through Liouville's theorem. Suppose, for contradiction, that a nonconstant polynomial p(z) had no root. Then 1 / p(z) would be holomorphic on the entire plane (the denominator never vanishes). As |z| grows large, |p(z)| grows without bound because the leading term dominates, so 1 / p(z) tends to 0; combined with continuity this makes 1 / p(z) bounded everywhere. A bounded entire function is constant by Liouville, so p would be constant too — contradicting our assumption. Hence p must have a root. Factoring it out and repeating gives all n roots.

The name is a little historical: the theorem is not really about algebra's foundations and every known proof leans on some analysis or topology (continuity, completeness, or winding numbers) — there is no purely algebraic proof. What complex analysis adds is elegance: the whole result drops out of the rigidity packaged in Liouville's theorem. It is a showpiece for how the analytic side of the subject settles a purely algebraic question.