fundamental theorem of algebra

The fundamental theorem of algebra is the promise that the complex numbers are “big enough.” It says that every polynomial equation of degree at least one, with complex coefficients, has at least one complex solution. No matter how nasty the polynomial, you never have to invent yet another new kind of number to solve it — the complex numbers already contain a root.

From this one statement a stronger fact follows by repeated factoring: a polynomial of degree n has exactly n complex roots when you count each root by how many times it repeats (its multiplicity). Equivalently, every such polynomial factors completely into linear pieces over the complex numbers. This is why mathematicians say the complex numbers are algebraically closed.

Two honest cautions. First, the theorem guarantees roots exist; it does not hand you a formula to find them, and for degree five and higher no general formula in radicals can exist. Second, the name is a bit misleading — every known proof relies on ideas from analysis or topology, not algebra alone.