University of Helmstedt1799
A new proof of the theorem that every integral rational algebraic function of one variable can be resolved . . .
Sometimes referred to as the Prince of Mathematicians, and greatest mathematician since antiquity, Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences".
In his 1799 doctorate in absentia, "A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree," Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work.
Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.