Fundamental Theorem of Algebra

Guarantees that the complex numbers are, in a precise sense, complete enough for algebra: no equation solvable in principle needs a still larger number system than the complex numbers to find its roots. Several mathematicians, including d'Alembert and Euler, attempted proofs earlier in the eighteenth century, but each had gaps; Carl Friedrich Gauss's 1799 doctoral dissertation gave the first substantially rigorous proof, and he returned to give three further, increasingly rigorous proofs over his career.